正单变量多项式的庞加莱乘子的柯蒂斯界的最优性
Optimality of Curtiss Bound on Poincare Multiplier for Positive Univariate Polynomials
AI总结：
本文证明对于正单变量多项式，在所有仅依赖于非实根辐角的乘子次数上界中，柯蒂斯界是最优的。
摘要：
设$f$为首一单变量多项式且常数项非零。若$f(x)$对所有$x\geq0$均为正，则称$f$为正的。若$f$的所有系数非负，则$f$平凡为正。1883年，庞加莱证明了$f$为正当且仅当存在首一多项式$g$使得$gf$的所有系数非负。这样的多项式$g$称为正多项式$f$的庞加莱乘子。当然，人们希望找到次数最小的乘子。这自然引发了一个挑战：寻找乘子最小次数的上界。1918年，柯蒂斯给出了这样的界。柯蒂斯还证明了当$f$的次数为1或2时，该界是最优的（最小的）。容易证明当$f$的次数更高时，该界并非最优。柯蒂斯界是一个仅依赖于$f$的非实根的辐角（幅角）的简单表达式。在本文中，我们证明了在所有仅依赖于辐角的界中，柯蒂斯界是最优的。
英文摘要：
Let $f$ be a monic univariate polynomial with non-zero constant term. We say that $f$ is positive if $f(x)$ is positive over all $x\geq0$. If all the coefficients of $f$ are non-negative, then $f$ is trivially positive. In 1883, Poincaré proved that$f$ is positive if and only if there exists a monic polynomial $g$ such that all the coefficients of $gf$ are non-negative. Such polynomial $g$ is called a Poincaré multiplier for the positive polynomial $f$. Of course one hopes to find a multiplier with smallest degree. This naturally raised a challenge: find an upper bound on the smallest degree of multipliers. In 1918, Curtiss provided such a bound. Curtiss also showed that the bound is optimal (smallest) when degree of $f$ is 1 or 2. It is easy to show that the bound is not optimal when degree of $f$ is higher. The Curtiss bound is a simple expression that depends only on the angle (argument) of non-real roots of $f$. In this paper, we show that the Curtiss bound is optimal among all the bounds that depends only on the angles.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
