diff --git "a/FdAyT4oBgHgl3EQfe_hY/content/tmp_files/load_file.txt" "b/FdAyT4oBgHgl3EQfe_hY/content/tmp_files/load_file.txt"
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf,len=1222
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='00331v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='AG]  1 Jan 2023  Optimality of Curtiss Bound on  Poincare Multiplier for  Positive Univariate Polynomials  Hoon Hong and Brittany Riggs  December 31 2022  Abstract  Let f be a monic univariate polynomial with non-zero constant term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We say that f is positive if  f(x) is positive over all x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' If all the coefficients of f are non-negative, then f is trivially positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  In 1888, Poincar´e proved thatf is positive if and only if there exists a monic polynomial g such that all  the coefficients of gf are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Such polynomial g is called a Poincar´e multiplier for the positive  polynomial f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Of course one hopes to find a multiplier with smallest degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' This naturally raised a  challenge: find an upper bound on the smallest degree of multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' In 1918, Curtiss provided such a  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Curtiss also showed that the bound is optimal (smallest) when degree of f is 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' It is easy to  show that the bound is not optimal when degree of f is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' The Curtiss bound is a simple expression  that depends only on the angle (argument) of non-real roots of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' In this paper, we show that the Curtiss  bound is optimal among all the bounds that depends only on the angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  1  Introduction  Let f be a monic univariate polynomial with non-zero constant term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We say that f is positive if f(x) is  positive over all x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Consider the following small (toy) examples,  f1 = x4 + x3 + 10x2 + 2x + 10  f2 = x4 − x3 − 10x2 − 2x + 10  f3 = x4 − x3 + 10x2 − 2x + 10  Note that all the coefficients of f1 are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Thus it is trivial to see that f1 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' However f2  and f3 have non-negative coefficients, and thus it is not obvious whether they are positive or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' It turns  out that f2 is not positive and f3 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  In 1888, Poincar´e [5] proved a general result that implies the following specific claim: f is positive if  and only if there exists a monic polynomial g such that all the coefficients of gf are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Such  polynomial g is called a Poincar´e multiplier for the positive polynomial f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' For the above examples, we have  Since f2 is not positive, there is no a Poincar´e multiplier for f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Since f3 is positive, there is a Poincar´e multiplier for f3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' For instance, let g = x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then  gf3 = (x + 1)  �  x4 − x3 + 10x2 − 2x + 10  �  = x5 + 9x3 + 8x2 + 8x + 10  Note that all the coefficients of gf3 are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  1  Note that there are (infinitely) many Poincar´e multipliers for f3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' For instance, let g = x2 + x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then  gf3 =  �  x2 + x + 1  � �  x4 − x3 + 10x2 − 2x + 10  �  = x6 + 10x4 + 7x3 + 18x2 + 8x + 10  Note that all the coefficients of gf3 are again non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Of course one hopes to find a Poincar´e multiplier  with optimal (smallest) degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' It turns out that the smallest degree for Poincar´e multipliers for f3 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  This naturally raised a challenge: find an upper bound on the smallest degree of multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  In 1918, Curtiss [3] provided such a bound (See Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Curtiss also showed that the bound is  optimal when degree of f  is 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' It is easy to show that the bound is not optimal when degree of f is  higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' For example, the Curtiss bound for f3 is 2, which is bigger than the optimal degree which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  It seems that Curtiss bound was forgotten for almost a century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' In 2010 Avenda˜no [1] (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1),  evidently not informed of the Curtiss bound, derived another bound implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Its derivation is very elegant  and short, but it turns out that the implied bound is roughly twice the Curtiss bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Boththe Curtiss bound and Avenda˜no bound depend only on the angles (arguments) of non-real roots of  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' The main contribution of this paper is to show that the Curtiss bound is optimal among all the bounds  that depends only on the angles of the non-real roots (see Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2  Main Result  Let f ∈ R [x] be monic such that ∀  x≥0f (x) > 0, that is, f does not have any non-negative real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Definition 1 (Optimal Bound) The optimal degree for f, written as opt (f), is defined by  opt (f) =  min  g∈R[x]\\0  coeff(gf)≥0  deg(g)  Theorem 1 (Curtiss Bound 1918[3]) Let r1e±iθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , rme±iθm be the non-real roots of f where multiple  roots are repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let  b (f) =  m  �  i=1  �� π  θi  �  − 2  �  Then opt (f) ≤ b (f) and the equality holds if deg f ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Theorem 2 (Main Result: Angle-Based Optimality of Curtiss Bound) We have  ∀  θ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',θm∈(0,π)  ∃  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',pt>0  ∃  r1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rm>0 opt (f) = b (f)  3  Proof of Curtiss Bound (Theorem 1)  In this section, we will provide an alternative proof for Curtiss’ theorem (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' This alternative proof  will be based on a new proof strategy that will be found crucial for proving our main result (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let f ∈ R [x] be monic without losing generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Assume that f does not have any non-negative real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  The problem is to find non-zero g ∈ R [x] such that coeffs (gf) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We will reduce the problem to that of  linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let  f = anxn + · · · + a0x0  g = bsxs + · · · + b0x0  where an = 1 and bs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We first rewrite them using vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let  a =  �  a0  · ·  an  �  b =  �  b0  · ·  bs  �  2  and let  xk =  \uf8ee  \uf8ef\uf8ef\uf8f0  x0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  xk  \uf8f9  \uf8fa\uf8fa\uf8fb  Then we can write f and g compactly as  f = axn  g = bxs  Let  As =  \uf8ee  \uf8ef\uf8ef\uf8f0  a0  · ·  · ·  an  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  a0  · ·  · ·  an  \uf8f9  \uf8fa\uf8fa\uf8fb ∈ R(s+1)×(s+n+1)  Lemma 3 coeffs (gf) = bAs  Proof: Note  gf = (bxs) (axn)  = b (xsaxn)  = b  \uf8ee  \uf8ef\uf8ef\uf8f0  x0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  xs  \uf8f9  \uf8fa\uf8fa\uf8fb  �  a0  · ·  an  �  \uf8ee  \uf8ef\uf8ef\uf8f0  x0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  xn  \uf8f9  \uf8fa\uf8fa\uf8fb  = b  \uf8ee  \uf8ef\uf8ef\uf8f0  a0  · ·  · ·  an  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  a0  · ·  · ·  an  \uf8f9  \uf8fa\uf8fa\uf8fb  \uf8ee  \uf8ef\uf8ef\uf8f0  x0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  xs+n  \uf8f9  \uf8fa\uf8fa\uf8fb  = bAsxs+n  Hence coeffs (gf) = bAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' □  We partition As into two submatrices as As = [Ls|Rs] where  Ls =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  a0  · ·  an−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  a0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ∈ R(s+1)×n  and Rs =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  an  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  a0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  a0  · ·  · ·  an  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ∈ R(s+1)×(s+1)  Let  c = bRs  ∈ R1×(s+1)  Ts = R−1  s Ls  ∈ R(s+1)×n  Lemma 4 coeffs (gf) = c [Ts|I]  3  Proof: Note  bAs = b  �  RsR−1  s  �  As  = (bRs)  �  R−1  s As  �  = (bRs)  �  R−1  s  [Ls|Rs]  �  = (bRs)  �  R−1  s Ls|R−1  s Rs  �  = (bRs)  �  R−1  s Ls|I  �  = c [Ts|I]  where c = bRs  and Ts = R−1  s Ls  □  Lemma 5 We have  ∃  g̸=0, deg(g)≤s coeffs (gf) ≥ 0  ⇐⇒  ConvexHull (Ts) ∩ Rn  ≥0 ̸= ∅  where Ts is a viewed as a set of row vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: Note  ∃  g̸=0, deg(g)≤s coeffs (gf) ≥ 0  ⇐⇒  ∃  c̸=0 c [Ts|I] ≥ 0  (from Lemma 4)  ⇐⇒  ∃  c̸=0 cTs ≥ 0 and c ≥ 0  ⇐⇒  ∃  c≥0, c̸=0 cTs ≥ 0  ⇐⇒  ∃  c��0, c0+···+cs=1 cTs ≥ 0  ⇐⇒  ConvexHull (Ts) ∩ Rn  ≥0 ̸= ∅  □  Let  f = an  n  �  i=1  (x − αi)  Lemma 6 The entries of Ts are given by  Tskℓ = −|N|  |D|  where  D =  \uf8ee  \uf8ef\uf8ef\uf8f0  α0  1  · ·  α0  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αn−1  1  · ·  αn−1  n  \uf8f9  \uf8fa\uf8fa\uf8fb  and N is obtained from D by replacing the ℓ-th row with  �  αk+n  1  · ·  αk+n  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: Note  xsf = Asxs+n  4  = RsR−1  s Asxs+n  = RsR−1  s  [Ls|Rs] xs+n  = Rs  �  R−1  s Ls|R−1  s Rs  �  xs+n  = Rs [Ts|I] xs+n  = Rs  \uf8eb  \uf8ec  \uf8ec  \uf8edTs  \uf8ee  \uf8ef\uf8ef\uf8f0  x0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  xn−1  \uf8f9  \uf8fa\uf8fa\uf8fb +  \uf8ee  \uf8ef\uf8ef\uf8f0  xn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  xs+n  \uf8f9  \uf8fa\uf8fa\uf8fb  \uf8f6  \uf8f7  \uf8f7  \uf8f8  By evaluating the above on each root, we have  \uf8ee  \uf8ef\uf8ef\uf8f0  α0  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αs  i  \uf8f9  \uf8fa\uf8fa\uf8fb f (αi) = Rs  \uf8eb  \uf8ec  \uf8ec  \uf8edTs  \uf8ee  \uf8ef\uf8ef\uf8f0  α0  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αn−1  i  \uf8f9  \uf8fa\uf8fa\uf8fb +  \uf8ee  \uf8ef\uf8ef\uf8f0  αn  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αs+n  i  \uf8f9  \uf8fa\uf8fa\uf8fb  \uf8f6  \uf8f7  \uf8f7  \uf8f8  Since f (αi) = 0, we have  0 = Rs  \uf8eb  \uf8ec  \uf8ec  \uf8edTs  \uf8ee  \uf8ef\uf8ef\uf8f0  α0  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αn−1  i  \uf8f9  \uf8fa\uf8fa\uf8fb +  \uf8ee  \uf8ef\uf8ef\uf8f0  αn  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αs+n  i  \uf8f9  \uf8fa\uf8fa\uf8fb  \uf8f6  \uf8f7  \uf8f7  \uf8f8  Since Rs is an invertible matrix, we have  0 = Ts  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  α0  i  α1  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αn−1  i  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  +  \uf8ee  \uf8ef\uf8ef\uf8f0  αn  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αs+n  i  \uf8f9  \uf8fa\uf8fa\uf8fb  Rearranging,  Ts  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  α0  i  α1  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αn−1  i  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  = −  \uf8ee  \uf8ef\uf8ef\uf8f0  αn  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αn+s  i  \uf8f9  \uf8fa\uf8fa\uf8fb  Combining the above equations for all the roots, we have  Ts  \uf8ee  \uf8ef\uf8ef\uf8f0  α0  1  · ·  α0  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αn−1  1  · ·  αn−1  n  \uf8f9  \uf8fa\uf8fa\uf8fb = −  \uf8ee  \uf8ef\uf8ef\uf8f0  αn  1  · ·  αn  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αs+n  1  · ·  αs+n  n  \uf8f9  \uf8fa\uf8fa\uf8fb  By applying Cramer’s rule, we have  Tskℓ = −|N|  |D|  where  D =  \uf8ee  \uf8ef\uf8ef\uf8f0  α0  1  · ·  α0  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  αn−1  1  · ·  αn−1  n  \uf8f9  \uf8fa\uf8fa\uf8fb  and N is obtained from D by replacing the ℓ-th row with  �  αk+n  1  · ·  αk+n  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' □  5  Remark 7 Note that Tskℓ does not depend on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Thus we will often write it as Tkℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Lemma 8 Let f ∈ R[x] be such that deg(f) = 2 without real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let the roots be α1 = reiθ and α2 = re−iθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Then we have  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Tk0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='+rk+2 sin (k + 1) θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='sin θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='r2 Im(αk+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=')  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Im(αi)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Tk1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='−rk+1 sin(k + 2)θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='sin θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='− Im(αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=')  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Im(αi)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Proof: From Lemma 6 we have  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Tk0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= −αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α2 − α1αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α2 − α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= −rk+2 +2i sin(k + 1) θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='−2i sinθ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= +rk+2 sin (k + 1) θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='sin θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= r2 Im(αk+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=')  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Im(αi)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Tk1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= −αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='− αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='α2 − α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= −rk+1 −2i sin(k + 2)θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='−2i sinθ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= −rk+1 sin(k + 2)θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='sin θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= − Im(αk+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=')  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Im(αi)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='□  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Lemma 9 Let f ∈ R[x] be such that deg(f) = 2 without real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let α1 = reiθ and α2 = re−iθ be the  roots of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  s =  �π  θ  �  − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Then  ∃  g̸=0, deg(g)≤s coeffs (gf) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: Note  ∃  g̸=0, deg(g)≤s coeffs (gf) ≥ 0  ⇐⇒  ConvexHull (Ts) ∩ Rn  ≥0 ̸= ∅  (from Lemma 5)  ⇐=  Ts0, Ts1 ≥ 0  ⇐⇒  sin (s + 1) θ ≥ 0 ∧ sin (s + 2) θ ≤ 0 (from Lemma 8)  ⇐=  0 < (s + 1)θ ≤ π ∧ π ≤ (s + 2)θ < 2π  ⇐⇒  s ≤ π  θ − 1 ∧ s ≥ π  θ − 2  ⇐⇒  π  θ − 2 ≤ s ≤ π  θ − 1  ⇐=  s =  �π  θ  �  − 2  □  Proof: [Proof of Theorem 1]  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let f be quadratic with non-real roots re±iθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note that bc(f) is optimal for any f with π  2 ≤ θ < π  since bc(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let 0 < θ < π  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let g be such that g ̸= 0 and deg(g) = v < s, where s = bc(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Thus  k  ≤ v  =⇒  k  < s  =⇒  k  <  �π  θ  �  − 2  =⇒  k  < π  θ − 2  (since k ≤  �π  θ  �  − 3)  =⇒  (k + 2)θ  < π  =⇒  sin(k + 2)θ  > 0  (since (k + 2)θ > 0)  =⇒  −rk+1 sin(k + 2)θ  sin θ  < 0  ⇐⇒  Tvk1  < 0  ⇐⇒  ConvexHull (Tv) ∩ R2  ≥0  = ∅  ⇐⇒  coeffs (gf)  < 0  (by Lemma 5)  Hence, opt(f) ̸< s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' By Lemma 9, opt(f) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Consider the factorization of f over R into linear and irreducible quadratic factors as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  f = (x + p1) · · · (x + pt)  �  x2 − 2r1 cos θ1x + r2  1  �  · ·  �  x2 − 2rm cos θmx + r2  m  �  f = l1 · · · lt q1 · · · qm  where  li = x + pi  qi = x2 − 2ri cos θix + r2  i  where again −pi stand for negative real roots and ri (cos θi ± i sin θi) stand for the complex conjugate  root pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' The coefficients of each li and qi with π  2 ≤ θi < π are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' From Lemma 8, for those  qi with 0 < θi < π  2 , there exists non-zero gi ∈ R[x] such that coeffs (gi qi) ≥ 0 and deg(gi) =  � π  θi  �  − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let g = g1 · · · gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then coeffs (gf) ≥ 0 and deg(g) =  m  �  i=1  �� π  θi  �  − 2  �  = b(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hence we have  opt (f) ≤ b (f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  □  4  Proof of Angle-Based Optimality (Theorem 2)  Let  f = fπ,p fφ,rφ fθ,rθ  fπ,p =  �  1≤i≤t  (x + pi) where pi > 0  fφ,rφ =  �  1≤i≤k  π  2 ≤φi<π  �  x2 − 2rφi cos φi x + r2  φi  �  where rφi > 0 and π > φ1 ≥ · · · ≥ φk ≥ π  2  7  fθ,rθ =  �  1≤i≤ℓ  0<θi< π  2  �  x2 − 2rθi cos θi x + r2  θi  �  where rθi > 0 and π  2 > θ1 ≥ · · · ≥ θℓ > 0  where k + ℓ = m (the number of complex root pairs of f) and 2m + t = n = deg(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: [Proof of Theorem 2]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We need to show  ∀  π>φ1≥···≥φk≥ π  2  ∀  π  2 >θ1≥···≥θℓ>0  ∃  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',pt>0  ∃  rφ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rφk>0  ∃  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt (f) = b (f)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let π > φ1 ≥ · · · ≥ φk ≥ π  2 and π  2 > θ1 ≥ · · · ≥ θℓ > 0 be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We need to show  ∃  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',pt>0  ∃  rφ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rφk>0  ∃  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt (f) = b (f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  We need to find a witness for p, rφ, rθ such that opt (f) = b (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We propose a witness candidate as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) From Lemma 10, for the fixed θ, we have  ∃  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt (fθ,rθ) = b (fθ,rθ)  (1)  (b) From Lemma 15, for the fixed φ and θ, we have  ∃  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',pt>0  ∃  rφ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rφk>0  ∀  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt  �  fπ,p fφ,rφ fθ,rθ  �  = opt (fθ,rθ)  (2)  (c) We propose p, rφ, rθ appearing in the above two facts as a witness candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We verify that the proposed candidate is indeed a witness, that is, opt (f) = b (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Note  opt (f) = opt  �  fπ,p fφ,rφ fθ,rθ  �  = opt (fθ,rθ)  by (2)  = b (fθ,rθ) by (1)  = b (fπ,p) + b  �  fφ,rφ  �  + b (fθ,rθ) since b (fπ,p) = b  �  fφ,rφ  �  = 0  = b  �  fπ,p fφ,rφ fθ,rθ  �  = b (f)  □  5  Supporting Lemmas for Proof of Theorem 2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1  Concerning Irreducible Quadratic Factors with 0 < θ < π  2  Let  αi = rieiθi  for 1 ≤ i ≤ ℓ  8  αm+i = rie−iθi  for 1 ≤ i ≤ ℓ  ti = cos θi  f =  ℓ  �  i=1  �  x2 − 2ritix + r2  i  �  =  ℓ  �  i=1  (x − αi)(x − αℓ+i) =  2ℓ  �  i=0  aixi  s = b(f)  g = xs−1 + bs−2xs−2 + · · · + b1x + b0  ck = coeff(gf, xk)  Note that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' ai = (−1)2ℓ−ie2ℓ−i (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) where ek (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) is the elementary symmetric polynomial of  degree k in the roots α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' When ℓ = 0, we define e0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' ti > 0 since 0 < θi < π  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Lemma 10  ∀  ℓ≥0  ∀  π  2 >θ1≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='≥θℓ>0  ∃  rθ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0 opt (fθ,rθ) = b (fθ,rθ)  Proof: We need to prove the following claim for every ℓ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ∀  π  2 >θ1≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='≥θℓ>0  ∃  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0 opt (fθ,rθ) = s  By Lemma 11, it suffices to show  ∀  π  2 >θ1≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='≥θℓ>0  ∃  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  ∀  g∈R[x]  deg(g)=s−1  ∃  0≤k≤2ℓ+s−1 ck < 0  ⇐⇒  ∀  0<t1≤···≤tℓ<1  ∃  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0 ∀  b  �  0≤k≤2ℓ+s−1  ck < 0  ⇐⇒  ∀  0<t1≤···≤tℓ<1  ∃  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0 ∀  b  �  0≤k≤2ℓ+s−2  ck < 0  (since the leading coefficient bs−1 = 1)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Case 1: ℓ = 0  Here, f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' The claim is trivially true since opt (fθ,rθ) = b (fθ,rθ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Case 2: ℓ = 1  Immediate from Remark ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='.  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Case 3: ℓ = 2  Immediate from Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Case 3: ℓ ≥ 3  Immediate from Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  □  Lemma 11  ∄  g, deg(g)=s ∀  k ck ≥ 0  =⇒  ∄  g, deg(g)<s ∀  k ck ≥ 0  Proof: We will prove via the contrapositive:  ∃  g, deg(g)<s ∀  k ck ≥ 0  =⇒  ∃  g, deg(g)=s ∀  k ck ≥ 0  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Assume  ∃  g, deg(g)<s  ∀  k  ck ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Then there exists a g with deg(g) = t < s such that gf has all  non-negative coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let u = s − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Consider the multiplier xu g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note that xu gf = xu(gf) must have all non-negative coefficients and  deg(xu g) = u + t = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then there exists a multiplier, xu g with degree equal to s such that the product has all non-negative  coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hence we have  ∃  g, deg(g)<s ∀  k ck ≥ 0  =⇒  ∃  g, deg(g)=s ∀  k ck ≥ 0  □  Lemma 12 For ℓ = 2,  ∀  0<t1≤t2<1  ∃  r1,r2>0 ∀  b  �  0≤k≤2ℓ+s−2  ck < 0  Proof: We will prove the following stronger statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ∀  0<t1≤t2<1  ∃  r1,r2>0 ∀  b  �  k∈{s−2,s−1, 2ℓ+s−3, 2ℓ+s−2}  ck < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let t1, t2 be such that 0 < t1 ≤ t2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let r1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note  cs−2 = a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + a4bs−6  = a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + bs−6  cs−1 = a0bs−1 + a1bs−2 + a2bs−3 + a3bs−4 + a4bs−5  = a0 + a1bs−2 + a2bs−3 + a3bs−4 + bs−5  c2ℓ+s−3 = a2bs−1 + a3bs−2 + a4bs−3  = a2 + a3bs−2 + bs−3  c2ℓ+s−2 = a3bs−1 + a4bs−2  = a3 + bs−2  (a) Note that coeff(gf, xk) =  �  i+j=k  aibj =  k  �  i=0  aibk−i for 0 ≤ k ≤ 2ℓ + s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Recall from Lemma 3 for 0 ≤ i ≤ 2ℓ and 0 ≤ j ≤ s − 1  coeffs(gf) =  �  b0  · ·  bs−1  �  \uf8ee  \uf8ef\uf8ef\uf8f0  a0  · ·  · ·  a2ℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  a0  · ·  · ·  a2ℓ  \uf8f9  \uf8fa\uf8fa\uf8fb  Then  coeff(gf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' xk) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i+j=k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='aibj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='(b) Then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='cs−2 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i+j=s−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='aibj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + a4bs−6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + bs−6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='since a4 = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='cs−1 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i+j=s−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='aibj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= a0bs−1 + a1bs−2 + a2bs−3 + a3bs−4 + a4bs−5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= a0 + a1bs−2 + a2bs−3 + a3bs−4 + bs−5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='since a4 = bs−1 = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='c2ℓ+s−3 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i+j=2ℓ+s−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='aibj =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i+j=s+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='aibj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= a2bs−1 + a3bs−2 + a4bs−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= a2 + a3bs−2 + bs−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='since a4 = bs−1 = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='c2ℓ+s−2 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i+j=2ℓ+s−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='aibj =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i+j=s+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='aibj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= a3bs−1 + a4bs−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='= a3 + bs−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='since a4 = bs−1 = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 1: When s = 2,  ∃  r(1)  2  >0  ∀  r2≥r(1)  2  ∀  b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Note  cs−1 = a0 + a1b0  c2ℓ+s−2 = a3 + b0  (b) We have  cs−1 < 0  ⇐⇒  b0 > −a0  a1  ⇐⇒  b0 > e4(α1, α2, α3, α4)  e3(α1, α2, α3, α4)  ⇐⇒  b0 >  r1r2  2r2t1 + 2r1t2  c2ℓ+s−2 < 0  ⇐⇒  b0 < −a3  ⇐⇒  b0 < e1(α1, α2, α3, α4)  ⇐⇒  b0 < 2r1t1 + 2r2t2  (c) Note  ∃  r2>0 ∀  b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note  ∀  b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0  ⇐=  r1r2  2r2t1 + 2r1t2  < 2r1t1 + 2r2t2  ⇐⇒  r1r2  (2r2t1 + 2r1t2)(2r1t1 + 2r2t2)  < 1  Note  ∀  b0  lim  r2→∞  r1r2  (2r2t1 + 2r1t2)(2r1t1 + 2r2t2) = 0  since  degr2 (r1r2) = 1  degr2 ((2r2t1 + 2r1t2)(2r1t1 + 2r2t2)) = 2  (d) Hence  ∃  r2>0 ∀  b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let s ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note  cs−2 = 0 ⇐⇒ bs−3 = −a0  a1  bs−2 − a2bs−4 + a3bs−5 + bs−6  a1  c2ℓ+s−3 = 0 ⇐⇒ bs−3 = −a3bs−2 − a2  Let b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , bs−4 be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let Lk be the line given by ck = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let µk be the slope of Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then µs−2 = −a0  a1  and µ2ℓ+s−3 = −a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Since ai = (−1)4−ie4−i (α1, α2, α3, α4), we have  a0 = e4 (α1, α2, α3, α4)  a1 = −e3 (α1, α2, α3, α4)  a3 = −e1 (α1, α2, α3, α4)  Then  µs−2  = e4 (α1, α2, α3, α4)  e3 (α1, α2, α3, α4)  =  r1r2  2r2t1 + 2r1t2  µ2ℓ+s−3  = e1 (α1, α2, α3, α4)  = 2r1t1 + 2r2t2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 2:  ∃  r(2)  2  >0  ∀  r2≥r(2)  2  µ2ℓ+s−3 > µs−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Note  µ2ℓ+s−3  > µs−2  ⇐⇒  2r1t1 + 2r2t2  >  r1r2  2r2t1 + 2r1t2  ⇐⇒  1  >  r1r2  (2r2t1 + 2r1t2)(2r1t1 + 2r2t2)  (b) Note  lim  r2→∞  r1r2  (2r2t1 + 2r1t2)(2r1t1 + 2r2t2) = 0  since  degr2 (r1r2) = 1  degr2 ((2r2t1 + 2r1t2)(2r1t1 + 2r2t2)) = 2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let r(2)  2  be such that  ∀  r2≥r(2)  2  µ2ℓ+s−3 > µs−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Such r(2)  2  exists due to the previous claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let r2 be  arbitrary but fixed such that r2 ≥ r(2)  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Over the space (bs−2, bs−3), there exists a unique intersection point of Ls−2 and L2ℓ+s−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let (p, q) be  the intersection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We have p = a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)  a0a4 − a1a3  and q = −a0a3 − a3(−a2bs−4 − a3bs−5 − bs−6)  a0a4 − a1a3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='cs−2 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='c2ℓ+s−3 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + bs−6 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2 + a3bs−2 + a4bs−3 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0bs−2 + a1bs−3 = −a2bs−4 − a3bs−5 − bs−6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a3bs−2 + a4bs−3 = −a2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='−a2bs−4 − a3bs−5 − bs−6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='−a3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−2 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='−a2bs−4 − a3bs−5 − bs−6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='−a3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−3 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='−a2bs−4 − a3bs−5 − bs−6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='−a3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−2 = a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0a4 − a1a3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−3 = −a0a3 − a3(−a2bs−4 − a3bs−5 − bs−6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0a4 − a1a3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 3:  ∃  r(2)  2  >0  ∀  r2≥r(2)  2  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2>p  cs−2 < 0 ∨ c2ℓ+s−3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let r(2)  2  be such that  ∀  r2≥r(2)  2  µ2ℓ+s−3 > µs−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' In the above, we have shown that such r(2)  2  exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let r2 ≥ r(2)  2  be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , bs−4 be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We need to show  ∀  bs−2,bs−3  bs−2>p  cs−2 < 0 ∨ c2ℓ+s−3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Over the space (bs−2, bs−3) where bs−2 > p, we have  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' c2ℓ+s−3 = 0 line and cs−2 = 0 line do not intersect  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' c2ℓ+s−3 = 0 line is above cs−2 = 0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (b) Let (bs−2, bs−3) be an arbitrary but fixed point such that bs−2 > p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then (bs−2, bs−3) is above  Ls−2 or below L2ℓ+s−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (c) Note  (bs−2, bs−3) is above Ls−2  ⇐⇒  bs−3 > − a0  a1 bs−2 − a2bs−4+a3bs−5+bs−6  a1  ⇐⇒  cs−2 < 0  (bs−2, bs−3) is below L2ℓ+s−3  ⇐⇒  bs−3 < −a3bs−2 − a2  ⇐⇒  c2ℓ+s−3 < 0  (d) Thus cs−2 < 0 or c2ℓ+s−3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 4:  ∃  r(3)  2  >0  ∀  r2≥r(3)  2  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2≤p  c2ℓ+s−2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Note  ∀  r2>0  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2≤p  a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)  e1 (α1, α2, α3, α4) (a0a4 − a1a3)  < 1  =⇒  c2ℓ+s−2 < 0  13  since  c2ℓ+s−2  < 0  ⇐⇒  bs−2  < −a3  ⇐=  p  < −a3  since bs−2 ≤ p  ⇐⇒  a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)  a0a4 − a1a3  < e1 (α1, α2, α3, α4)  ⇐⇒  a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)  e1 (α1, α2, α3, α4) (a0a4 − a1a3)  < 1  (b) Let ek = ek (α1, α2, α3, α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note  a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)  e1 (α1, α2, α3, α4) (a0a4 − a1a3)  = e3e1 − e0(e2bs−4 − e1bs−5 + bs−6)  e1 (e4e0 − e3e1)  (c) Note  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  lim  r2→∞  e3e1 − e0(e2bs−4 − e1bs−5 + bs−6)  e1 (e4e0 − e3e1)  = 0  since  degr2  �e3e1 − e0(e2bs−4 − e1bs−5 + bs−6)  e1 (e4e0 − e3e1)  �  ≤ 3  degr2 (e1 (e4e0 − e3e1)) = 4  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 5:  ∃  r2>0  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  �  0≤k≤2ℓ+s−2  ck < 0  From Claim 1, when s = 2, for some r(1)  2  > 0 we have  ∃  r2>0 ∀  b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  From Claim 3, when s ≥ 3, for some r(2)  2  > 0 we have  ∀  r2>r(1)  2  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2>p  cs−2 < 0 ∨ c2ℓ+s−3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  From Claim 4, when s ≥ 3, for some r(3)  2  > 0 we have  ∀  r2≥r(2)  2  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2≤p  c2ℓ+s−2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Then for r∗  2 = max{r(1)  2 , r(2)  2 , r(3)  2 }, we have  ∀  r2≥r∗  2  \uf8ee  \uf8ef\uf8f0  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2>p  cs−1 < 0 ∨ cs−2 < 0 ∨ c2ℓ+s−3 < 0  ∧  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2≤p  cs−1 < 0 ∨ c2ℓ+s−2 < 0  \uf8f9  \uf8fa\uf8fb  Hence  ∃  r2>0  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  �  k∈{s−2,s−1,2ℓ+s−3,2+s−2}  ck < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  □  Lemma 13 For ℓ ≥ 3,  ∀  0<t1≤···≤tℓ<1  ∃  r1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rℓ>0 ∀  b  �  0≤k≤2ℓ+s−2  ck < 0  14  Proof: We will prove the following stronger statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ∀  0<t1≤···≤tℓ<1  ∃  r1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rℓ−1>0  C(r)<1  ∃  rℓ>0 ∀  b  �  k∈{s−2, 2ℓ+s−5, 2ℓ+s−2}  ck < 0  where C(r) = e0 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1) e2ℓ−2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1) e2ℓ−3 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , tℓ be such that 0 < t1 ≤ · · · ≤ tℓ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 1:  ∀  0<t1≤···≤tℓ−1<1  ∃  r1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rℓ−1>0 C(r) < 1  (a) Note  e0 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1) = 1  e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1) = α1 + · · · + αℓ−1 + αℓ+1 + · · · + α2ℓ−1  = (α1 + αℓ+1) + · · · + (αℓ−1 + α2ℓ−1)  = 2r1t1 + · · · + 2rℓ−1tℓ−1  e2ℓ−3 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1) =  �  i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',ℓ−1,ℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',2ℓ−1}  \uf8eb  \uf8ed�  j̸=i  αj  \uf8f6  \uf8f8  =  �  1≤i≤ℓ−1  \uf8eb  \uf8ed�  j̸=i  αj +  �  j̸=ℓ+i  αj  \uf8f6  \uf8f8  =  �  1≤i≤ℓ−1  \uf8eb  \uf8ed(αℓ+i + αi)  �  j̸=i,ℓ+1  αj  \uf8f6  \uf8f8  =  �  1≤i≤ℓ−1  \uf8eb  \uf8ed(2riti)  �  j̸=i  r2  j  \uf8f6  \uf8f8  e2ℓ−2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1) = α1 · · · αℓ−1αℓ+1 · · · α2ℓ−1  = α1αℓ+1 · · · αℓ−1α2ℓ−1  = r2  1 · · · r2  ℓ−1  Then  C(r) = e0 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1) e2ℓ−2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1) e2ℓ−3 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  =  r2  1 · · · r2  ℓ−1  (2r1t1 + · · · + 2rℓ−1tℓ−1)  ��  1≤i≤ℓ−1  �  (2riti) �  j̸=i r2  j  ��  =  r1 · · · rℓ−1  (2r1t1 + · · · + 2rℓ−1tℓ−1)  ��  1≤i≤ℓ−1  �  (2ti) �  j̸=i rj  ��  =  1  (2r1t1 + · · · + 2rℓ−1tℓ−1)  �2t1  r1  + · · · + 2tℓ−1  rℓ−1  �  (b) Consider the following witness candidates for the existentially quantified variables r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , rℓ−1:  r1 = 2t1  15  ri = 1  2ti  for 2 ≤ t ≤ ℓ − 1  (c) Obviously, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , rℓ−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note  1  (2r1t1 + 2r2t2 + · · · + 2rℓ−1tℓ−1)  �2t1  r1  + 2t2  r2  + · · · + 2tℓ−1  rℓ−1  �  =  1  �  2 (2t1) t1 + 2  � 1  2t2  �  t2 + · · · + 2  �  1  2tℓ−1  �  tℓ−1  � �  2t1  2t1  + 2t2  1  2t2  + · · · + 2tℓ−1  1  2tℓ−1  �  =  1  (4t2  1 + 1 + · · · + 1)  �  1 + 4t2  2 + · · · + 4t2  ℓ−1  �  < 1  Hence the candidates are witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , rℓ−1 > 0 be such that C(r) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note  cs−2 = a0bs−2 + a1bs−3 +  s−2  �  i=2  aib(s−2)−i  c2ℓ+s−5 = a2ℓ−4bs−1 + a2ℓ−3bs−2 + a2ℓ−2bs−3 + a2ℓ−1bs−4 + a2ℓbs−5  = a2ℓ−4 + a2ℓ−3bs−2 + a2ℓ−2bs−3 + a2ℓ−1bs−4 + bs−5  c2ℓ+s−2 = a2ℓ−1bs−1 + a2ℓbs−2  = a2ℓ−1 + bs−2  (a) Note that coeff(gf, xk) =  �  i+j=k  aibj =  k  �  i=0  aibk−i for 0 ≤ k ≤ 2ℓ + s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Recall from Lemma 3 for 0 ≤ i ≤ 2ℓ and 0 ≤ j ≤ s − 1  coeffs(gf) =  �  b0  · ·  bs−1  �  \uf8ee  \uf8ef\uf8ef\uf8f0  a0  · ·  · ·  a2ℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  a0  · ·  · ·  a2ℓ  \uf8f9  \uf8fa\uf8fa\uf8fb  Then  coeff(gf, xk) =  �  i+j=k  aibj  =  k  �  i=0  aibk−i  (b) Then  cs−2 =  s−2  �  i=0  aib(s−2)−i  16  = a0bs−2 + a1bs−3 +  s−2  �  i=2  aib(s−2)−i  c2ℓ+s−5 =  �  i+j=2ℓ+s−5  aibj  = a2ℓ−4bs−1 + a2ℓ−3bs−2 + a2ℓ−2bs−3 + a2ℓ−1bs−4 + a2ℓbs−5  = a2ℓ−4 + a2m−3bs−2 + a2m−2bs−3 + a2m−1bs−4 + bs−5  since a2ℓ = bs−1 = 1  c2���+s−2 =  �  i+j=2ℓ+s−2  aibj  = a2ℓ−1bs−1 + a2ℓbs−2  = a2ℓ−1 + bs−2  since a2ℓ = bs−1 = 1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note  cs−2 = 0 ⇐⇒ bs−3 = −a0  a1  bs−2 −  s−2  �  i=2  ai  a1  b(s−2)−i  c2ℓ+s−5 = 0 ⇐⇒ bs−3 = −a2ℓ−3  a2ℓ−2  bs−2 − a2ℓ−1bs−4 + bs−5 + a2ℓ−4  a2ℓ−2  Let b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , bs−4 be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let Lk be the line given by ck = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let µk be the slope of Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then µs−2 = −a0  a1  and µ2ℓ+s−5 = −a2ℓ−3  a2ℓ−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Since ai = (−1)2ℓ−ie2ℓ−i (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ), we have  a0 = e2ℓ (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  a1 = −e2ℓ−1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  a2ℓ−3 = −e3 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  a2ℓ−2 = e2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  Then µs−2 =  e2ℓ (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  e2ℓ−1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) and µ2ℓ+s−5 = e3 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  e2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 2:  ∃  r(1)  ℓ  >0  ∀  rℓ≥r(1)  ℓ  µ2ℓ+s−5 > µs−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Note as rℓ → ∞  e2ℓ (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) → αℓα2ℓ e2ℓ−2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  e2ℓ−1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) → αℓα2ℓ e2ℓ−3 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  e3 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) → αℓα2ℓ e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  e2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) → αℓα2ℓ e0 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  Then  µs−2 → e2ℓ−2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  e2ℓ−3 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  µ2ℓ+s−5 → e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  e0 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  17  (b) Then for sufficiently large rℓ,  µ2ℓ+s−5  > µs−2  ⇐⇒  e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  e0 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  > e2ℓ−2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  e2ℓ−3 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , αℓ−1, αℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ−1)  ⇐⇒  1  > C(r)  which is true for the r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , rℓ−1 we have chosen based on Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let r(1)  ℓ  be such that  ∀  rℓ≥r(1)  ℓ  µ2ℓ+s−5 > µs−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Such r(1)  ℓ  exists due to the previous claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let rℓ be  arbitrary but fixed such that rℓ ≥ r(1)  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Over the space (bs−2, bs−3), there exists a unique intersection point of Ls−2 and L2ℓ+s−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let (p, q) be  the intersection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We have p =  a2ℓ−2d0 − a1d1  a0a2ℓ−2 − a1a2ℓ−3  and q =  a0d1 − a2ℓ−3d0  a0a2ℓ−2 − a1a2ℓ−3  where d0 = −  s−2  �  i=2  aib(s−2)−i and d1 =  −a2ℓ−1bs−4 − bs−5 − a2ℓ−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='cs−2 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='c2ℓ+s−5 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0bs−2 + a1bs−3 + �s−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i=2 aib(s−2)−i = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−4 + a2ℓ−3bs−2 + a2ℓ−2bs−3 + a2ℓ−1bs−4 + bs−5 = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0bs−2 + a1bs−3 = − �s−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i=2 aib(s−2)−i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−3bs−2 + a2ℓ−2bs−3 = −a2ℓ−1bs−4 − bs−5 − a2ℓ−4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0bs−2 + a1bs−3 = d0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='where d0 = − �s−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='i=2 aib(s−2)−i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−3bs−2 + a2ℓ−2bs−3 = d1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='where d1 = −a2ℓ−1bs−4 − bs−5 − a2ℓ−4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='d0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='d1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−2 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='d0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='d1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−3 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='d0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='d1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='�������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='⇐⇒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−2 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a2ℓ−2d0 − a1d1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0a2ℓ−2 − a1a2ℓ−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='∧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='bs−3 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0d1 − a2ℓ−3d0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='a0a2ℓ−2 − a1a2ℓ−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 3:  ∃  r(1)  ℓ  >0  ∀  rℓ≥r(1)  ℓ  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2>p  cs−2 < 0 ∨ c2ℓ+s−5 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let r(1)  ℓ  be such that  ∀  rℓ≥r(1)  ℓ  µ2ℓ+s−5 > µs−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' In the above, we have shown that such r(1)  ℓ  exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let rℓ ≥ r(1)  ℓ  be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , bs−4 be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We need to show  ∀  bs−2,bs−3  bs−2>p  cs−2 < 0 ∨ c2ℓ+s−5 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Over the space (bs−2, bs−3) where bs−2 > p, we have  18  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' c2ℓ+s−5 = 0 line and cs−2 = 0 line do not intersect  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' c2ℓ+s−5 = 0 line is above cs−2 = 0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (b) Let (bs−2, bs−3) be an arbitrary but fixed point such that bs−2 > p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then (bs−2, bs−3) is above  Ls−2 or below L2ℓ+s−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (c) Note  (bs−2, bs−3) is above Ls−2  ⇐⇒  bs−3 > − a0  a1 bs−2 − d0  a1  ⇐⇒  cs−2 < 0  (bs−2, bs−3) is below L2ℓ+s−5  ⇐⇒  bs−3 < − a2ℓ−3  a2ℓ−2 bs−2 −  d1  a2ℓ−2  ⇐⇒  c2ℓ+s−5 < 0  since a1 = (−1)2ℓ−1e2ℓ−1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) < 0 and a2ℓ−2 = (−1)2e2 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (d) Thus cs−2 < 0 or c2ℓ+s−5 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 4:  ∃  r(2)  ℓ  >0  ∀  rℓ≥r(2)  ℓ  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2≤p  c2ℓ+s−2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Note  ∀  rℓ>0  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2≤p  a2ℓ−2d0 − a1d1  e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) (a0a2ℓ−2 − a1a2ℓ−3) < 1  =⇒  c2+s−2 < 0  since  c2ℓ+s−2  < 0  ⇐⇒  bs−2  < −a2ℓ−1  ⇐=  p  < −a2ℓ−1  since bs−2 ≤ p  ⇐⇒  a2ℓ−2d0 − a1d1  a0a2ℓ−2 − a1a2ℓ−3  < e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  ⇐⇒  a2ℓ−2d0 − a1d1  e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) (a0a2ℓ−2 − a1a2ℓ−3)  < 1  (b) Let ek = ek (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note  a2ℓ−2d0 − a1d1  e1 (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) (a0a2ℓ−2 − a1a2ℓ−3)  = −e2  �s−2  i=2 (−1)2ℓ−ie2ℓ−ib(s−2)−i + e2ℓ−1 (e1bs−4 − bs−5 − e4)  e1 (e2ℓe2 − e2ℓ−1e3)  (c) Note  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  lim  rℓ→∞  −e2  �s−2  i=2 (−1)2ℓ−ie2ℓ−ib(s−2)−i + e2ℓ−1 (e1bs−4 − bs−5 − e4)  e1 (e2ℓe2 − e2ℓ−1e3)  = 0  since  degrℓ  �  −e2  s−2  �  i=2  (−1)2ℓ−ie2ℓ−ib(s−2)−i + e2ℓ−1 (e1bs−4 − bs−5 − e4)  �  ≤ 4  degrℓ (e1 (e2ℓe2 − e2ℓ−1e3)) = 5  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Claim 5:  ∃  rℓ>0  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  �  0≤k≤2ℓ+s−2  ck < 0  From Claim 3, for some r(1)  ℓ  > 0 we have  ∀  rℓ>r(1)  ℓ  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2>p  cs−2 < 0 ∨ c2ℓ+s−5 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  19  From Claim 4, for some r(2)  ℓ  > 0 we have  ∀  rℓ≥r(2)  ℓ  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2≤p  c2ℓ+s−2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Then for r∗  ℓ = max{r(1)  ℓ , r(2)  ℓ }, we have  ∀  rℓ≥r∗  ℓ  \uf8ee  \uf8ef\uf8f0  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2>p  cs−2 < 0 ∨ c2ℓ+s−5 < 0  ∧  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  bs−2≤p  c2ℓ+s−2 < 0  \uf8f9  \uf8fa\uf8fb  Hence  ∃  rℓ>0  ∀  b0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',bs−2  �  k∈{s−2,2ℓ+s−5,2ℓ+s−2}  ck < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  □  Lemma 14 If arg(αi) < π  2 , then ek (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) > 0 for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: We will induct on ℓ, the number of quadratic factors of f with non-real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Base Case: ℓ = 0  Note that e0 = 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hypothesis: Assume ek (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ) > 0 for ℓ ≥ 1 and 0 ≤ k ≤ 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Induction Step: Prove ek  �  α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2(ℓ+1)  �  > 0 for 0 ≤ k ≤ 2(ℓ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Let fℓ =  ℓ  �  i=1  �  x2 − 2ritix + r2  i  �  and aℓ,k = coeff(fℓ, xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note that  fℓ+1  =  �  x2 − 2rℓ+1tℓ+1x + r2  ℓ+1  �  fℓ  aℓ+1,k  = aℓ,k−2 − 2rℓ+1,t+1aℓ,k−1 + r2  ℓ+1aℓ,k  Then  aℓ+1,k = (−1)2(ℓ+1)−ke2(ℓ+1)−k  �  α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2(ℓ+1)  �  = (−1)2ℓ−(k−2)e2ℓ−(k−2) (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  − 2rℓ+1tℓ+1(−1)2ℓ−(k−1)e2ℓ−(k−1) (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  + r2  ℓ+1(−1)2ℓ−ke2ℓ−k (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  Hence  e2(ℓ+1)−k  �  α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2(ℓ+1)  �  = e2ℓ−(k−2) (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  − 2rℓ+1tℓ+1(−1)−1e2ℓ−(k−1) (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  + r2  ℓ+1(−1)−2e2ℓ−k (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  = e2ℓ−(k−2) (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  + 2rℓ+1tℓ+1e2ℓ−(k−1) (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  + r2  ℓ+1e2ℓ−k (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , α2ℓ)  > 0  by the inductive hypothesis and the fact that rℓ+1, tℓ+1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  □  20  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2  Concerning Linear and Irreducible Quadratic Factors with π  2 < θ < π  Lemma 15 We have  ∀  π>φ1≥···≥φk≥ π  2  ∀  π  2 >θ1≥···≥θℓ>0  ∃  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',pt>0  ∃  rφ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rφk>0  ∀  rθ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',θℓ>0  opt  �  fπ,p fφ,rφ fθ,rθ  �  = opt (fθ,rθ)  Proof: We will first induct on k, the number of quadratic factors with π  2 ≤ φi < π in fφ,rφ, to show that  opt  �  fφ,rφ fθ,rθ  �  = opt (fθ,rθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Base: The claim holds for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hypo: Assume the claim holds for k quadratic factors with π  2 ≤ φi < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ∃  rφ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rφk>0  ∀  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt  �  fφ,rφ fθ,rθ  �  = opt (fθ,rθ)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Induction Step: Consider k + 1 quadratic factors with π  2 ≤ φi < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Assume rφ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , rφk > 0 are such that the induction hypothesis holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' By Lemma 19,  ∃  rφk+1 >0  ∀  rθ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt  ��  x2 − 2rφk+1 cos φk+1 x + r2  φk+1  �  fφ,rφ fθ,rθ  �  = opt  �  x2 − 2rφk+1 cos φk+1 x + r2  φk+1  �  + opt  �  fφ,rφ fθ,rθ  �  By Remark ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=', opt  �  x2 − 2rφk+1 cos φk+1 x + r2  φk+1  �  = 0, so we have  opt  ��  x2 − 2rφk+1 cos φk+1 x + r2  φk+1  �  fφ,rφ fθ,rθ  �  = opt  �  fφ,rφ fθ,rθ  �  = opt (fθ,rθ)  Hence, we have  ∃  rφ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rφk>0  ∀  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt  �  fφ,rφ fθ,rθ  �  = opt (fθ,rθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Now we will induct on t, the number of linear factors in fπ,p, to show that  ∃  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',pt>0  ∃  rφ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rφk>0  ∀  rθ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt  �  fπ,p fφ,rφ fθ,rθ  �  = opt (fθ,rθ)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Base: The claim holds for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hypo: Assume the claim holds for t linear factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ∃  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',pt>0  ∃  rφ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rφk>0  ∀  rθ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt  �  fπ,p fφ,rφ fθ,rθ  �  = opt (fθ,rθ)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Induction Step: Consider t + 1 linear factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Assume p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , pt, rφ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , rφk > 0 are such that the induction hypothesis holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' By Lemma 18,  ∃  pt+1>0  ∀  rθ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt ((x + pt+1) fπ,p fφ,σ fθ,τ) = opt (x + pt+1) + opt (fπ,p fφ,σ fθ,τ)  By Lemma 16, opt (x + pt+1) = 0, so we have  opt ((x + pt+1) fπ,p fφ,σ fθ,τ) = opt  �  fπ,p fφ,rφ fθ,rθ  �  = opt (fθ,rθ)  Hence, we have  ∃  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',pt>0  ∃  rφ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rφk>0  ∀  rθ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',rθℓ>0  opt  �  fπ,p fφ,rφ fθ,rθ  �  = opt (fθ,rθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  21  □  Notation 2 Note  hr = x + r  hθ,r = x2 − 2r cos θ x + r2  P = {f ∈ R[x] : f(x) > 0 for x ≥ 0}  Lemma 16  ∀  f∈P, deg(f)=1 opt (f) = b (f)  Proof: Note that b (f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' □  Lemma 17  ∀  f∈P, deg(f)=2 opt (f) = b (f)  Proof: By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' □  Lemma 18  ∀  f∈P, deg(f)≥1 ∃  r>0 opt (hr f) = opt (hr) + opt (f)  Proof: Let f ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Let h = hr f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We need to show ∃  r>0 opt(h) = opt(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We will divide the proof into several  claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  C1:  ∃  r>0 opt(h) = opt(f)  ⇐=  ∃  r>0 CH(T ∗  h,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , T ∗  h,s−1) ∩ Rn  ≥0 = ∅  where  n = deg (f)  s = opt(f)  The Th,i are the rows of Ts−1 for h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  T ∗  h,i is obtained from Th,i by deleting the first element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: Note  opt(h) = opt(f)  ⇐⇒  opt(h) ≥ s  since  opt(h) ≤ opt(f) + opt(x + r) = opt(f) = s  ⇐⇒  ¬  ∃  g̸=0, deg(g)<s coeffs(gh) ≥ 0  ⇐⇒  ¬  �  CH(Th,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , Th,s−1) ∩ Rn+1  ≥0  ̸= ∅  �  ⇐⇒  CH(Th,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , Th,s−1) ∩ Rn+1  ≥0  = ∅  ⇐=  CH(T ∗  h,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , T ∗  h,s−1) ∩ Rn  ≥0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  C2:  ∃  r>0 opt(h) = opt(f)  ⇐=  ∃  r>0εh (r) > 0  where εh (r) stands for the minimum Euclidean distance between CH(T ∗  h,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , T ∗  h,s−1) and Rn  ≥0, that  is,  εh(r) :=  min  x∈CH(T ∗  h,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',T ∗  h,s−1)  y∈Rn  ≥0  ∥x − y∥  Proof: Immediate from the above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  22  C3: εh (r) is continuous at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: We will divide it into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Note  εh(r) =  min  c∈Rs  ≥0  c0+···+cs−1=1  y∈Rn  ≥0  �����  s−1  �  i=0  ciT ∗  h,i − y  �����  =  min  c∈Rs  ≥0  c0+···+cs−1=1  y∈Rn  ≥0  �����  �s−1  �  i=0  ciT ∗  h,i,1 − y1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' ,  s−1  �  i=0  ciT ∗  h,i,n − yn  ������  =  min  z∈Rs+n  ≥0  z0+···+zs−1=1  �����  �s−1  �  i=0  ziT ∗  h,i,1 − zs , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' ,  s−1  �  i=0  ziT ∗  h,i,n − zs+n−1  ������  where z = (c, y)  = min  z∈C p (z, r)  where p(z, r) is Euclidean distance and  C =  �  z ∈ Rs+n  ≥0 : z1 + · · · + zs = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (b) Note, since p(z, r) is the distance between two closed sets, the minimum distance is realized by a  point in each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hence,  εh(r) = min  z∈C p (z, r) = inf  z∈C p (z, r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (c) Note the following:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' By Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='5 of [2], p(z, r) is a convex function since p(z, r) is a norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' By Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='5 of [2], εh(r) = inf  z∈C p (z, r) is a convex function, since C is convex and p(z, r)  is bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='3 in [4], since εh(r) is a convex function defined on a convex set R, εh(r) is  continuous on the relative interior of R, ri (R) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hence, εh(r) is continuous at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  C4: εh (0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: Let r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then h = x · f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Subclaim: T ∗  h,i,j = Tf,i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note that these are the entries of T ∗  h,s−1 and Tf,s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We will prove the  claim using the fact that Ts−1 = R−1  s−1Ls−1 for any f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note Rf,s−1 = Rh,s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' This is clear from the definition of Rs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hence R−1  f,s−1 = R−1  h,s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note that Lf,i,j = Lh,i,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' This is clear from the definition of Ls−1, since Lh,s−1 is composed  of Lf,s−1 with a column of zeroes added on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note that for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , s − 1 and j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , n − 1,  T ∗  h,i,j = Th,i,j+1  =  s−1  �  k=0  R−1  h,i,kLh,k,j+1  23  =  s−1  �  k=0  R−1  f,i,kLf,k,j  = Tf,i,j  Hence T ∗  h,i,j = Tf,i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (b) Subclaim: εh(0) = εf  where εf stands for the minimum Euclidean distance between CH(Tf,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , Tf,s−1) and Rn  ≥0, that  is,  εf :=  min  x∈CH(Tf,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',Tf,s−1)  y∈Rn  ≥0  ∥x − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  To see this, note  εh(0) =  min  c∈Rs  ≥0  c0+···+cs−1=1  y∈Rn  ≥0  �����  s−1  �  i=0  ciT ∗  h,i − y  �����  =  min  c∈Rs  ≥0  c0+···+cs−1=1  y∈Rn  ≥0  �����  s−1  �  i=0  ciTf,i − y  �����  =  min  x∈CH(Tf,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',Tf,s−1)  y∈Rn  ≥0  ∥x − y∥  = εf  (c) Subclaim: εf > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Note since opt (f) = s, we have CH(Tf,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , Tf,s−1) ∩ Rn  ≥0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Thus εf > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (d) From the above two subclaims, we have εh (0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  From the above four claims, we immediately have  ∃  r>0 opt(h) = opt(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  □  Lemma 19  ∀  f∈P, deg(f)≥1  ∀  π  2 ≤θ< π  1  ∃  r>0 opt (hθ,r f) = opt (hθ,r) + opt (f)  Proof: Let f ∈ P and h = hθ,r f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We need to show ∃  r>0 opt(h) = opt(f) since opt(hθ,r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We will divide  the proof into several claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  C1:  ∃  r>0 opt(h) = opt(f)  ⇐=  ∃  r>0 CH(T ∗  h,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , T ∗  h,s−1) ∩ Rn  ≥0 = ∅  where  n = deg (f)  s = opt(f)  The Th,i are the rows of Ts−1 for h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  T ∗  h,i is obtained from Th,i by deleting the first two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  24  Proof: Note  opt(h) = opt(f)  ⇐⇒  opt(h) ≥ s  since  opt(h) ≤ opt(f) + opt(hr,θ) = opt(f) = s  ⇐⇒  ¬  ∃  g̸=0, deg(g)<s coeffs(gh) ≥ 0  ⇐⇒  ¬  �  CH(Th,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , Th,s−1) ∩ Rn+2  ≥0  ̸= ∅  �  ⇐⇒  CH(Th,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , Th,s−1) ∩ Rn+2  ≥0  = ∅  ⇐=  CH(T ∗  h,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , T ∗  h,s−1) ∩ Rn  ≥0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  C2:  ∃  r>0 opt(h) = opt(f)  ⇐=  ∃  r>0εh (r) > 0  where εh (r) stands for the minimum Euclidean distance between CH(T ∗  h,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , T ∗  h,s−1) and Rn  ≥0, that  is,  εh(r) :=  min  x∈CH(T ∗  h,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',T ∗  h,s−1)  y∈Rn  ≥0  ∥x − y∥  Proof: Immediate from the above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  C3: εh (r) is continuous at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: We will divide it into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Note  εh(r) =  min  c∈Rs  ≥0  c0+···+cs−1=1  y∈Rn  ≥0  �����  s−1  �  i=0  ciT ∗  h,i − y  �����  =  min  c∈Rs  ≥0  c0+···+cs−1=1  y∈Rn  ≥0  �����  �s−1  �  i=0  ciT ∗  h,i,1 − y1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' ,  s−1  �  i=0  ciT ∗  h,i,n − yn  ������  =  min  z∈Rs+n  ≥0  z0+···+zs−1=1  �����  �s−1  �  i=0  ziT ∗  h,i,1 − zs , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' ,  s−1  �  i=0  ziT ∗  h,i,n − zs+n−1  ������  where z = (c, y)  = min  z∈C p (z, r)  where p(z, r) is Euclidean distance and  C =  �  z ∈ Rs+n  ≥0 : z1 + · · · + zs = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (b) Note, since p(z, r) is the distance between two closed sets, the minimum distance is realized by a  point in each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hence,  εh(r) = min  z∈C p (z, r) = inf  z∈C p (z, r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (c) Note the following:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' By Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='5 of [2], p(z, r) is a convex function since p(z, r) is a norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' By Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='5 of [2], εh(r) = inf  z∈C p (z, r) is a convex function, since C is convex and p(z, r)  is bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  25  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='3 in [4], since εh(r) is a convex function defined on a convex set R, εh(r) is  continuous on the relative interior of R, ri (R) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hence, εh(r) is continuous at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  C4: εh (0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Proof: Let r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Then h = x2 · f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (a) Subclaim: T ∗  h,i,j = Tf,i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note that these are the entries of T ∗  h,s−1 and Tf,s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' We will prove the  claim using the fact that Ts−1 = R−1  s−1Ls−1 for any f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note Rf,s−1 = Rh,s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' This is clear from the definition of Rs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Hence R−1  f,s−1 = R−1  h,s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note that Lf,i,j = Lh,i,j+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' This is clear from the definition of Ls−1, since Lh,s−1 is composed  of Lf,s−1 with two columns of zeroes added on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Note that for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , s − 1 and j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , n − 1,  T ∗  h,i,j = Th,i,j+2  =  s−1  �  k=0  R−1  h,i,kLh,k,j+2  =  s−1  �  k=0  R−1  f,i,kLf,k,j  = Tf,i,j  Hence T ∗  h,i,j = Tf,i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (b) Subclaim: εh(0) = εf  where εf stands for the minimum Euclidean distance between CH(Tf,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , Tf,s−1) and Rn  ≥0, that  is,  εf :=  min  x∈CH(Tf,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',Tf,s−1)  y∈Rn  ≥0  ∥x − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  To see this, note  εh(0) =  min  c∈Rs  ≥0  c0+···+cs−1=1  y∈Rn  ≥0  �����  s−1  �  i=0  ciT ∗  h,i − y  �����  =  min  c∈Rs  ≥0  c0+···+cs−1=1  y∈Rn  ≥0  �����  s−1  �  i=0  ciTf,i − y  �����  =  min  x∈CH(Tf,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=',Tf,s−1)  y∈Rn  ≥0  ∥x − y∥  = εf  (c) Subclaim: εf > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  Note since opt (f) = s, we have CH(Tf,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' , Tf,s−1) ∩ Rn  ≥0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Thus εf > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  (d) From the above two subclaims, we have εh (0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  From the above four claims, we immediately have  ∃  r>0 opt(h) = opt(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  □  26  References  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Avendano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Descartes’ rule of signs is exact!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Journal of Algebra - J ALGEBRA, 324:2884–2892, 11  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Boyd and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Vandenberghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Cambridge University Press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Curtiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Recent extensions of descartes rule of signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Annals of Mathematics, 19(4):251–278, 6 1918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Niculescu and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Persson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Convex functions and their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Springer Verlag New York, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Poincar´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Sur les ´equations alg´ebriques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content=' Comptes Rendus, 97:1418–1419, 1888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}
+page_content='  27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfe_hY/content/2301.00331v1.pdf'}