多元稳定多项式：理论与应用
Multivariate stable polynomials: theory and applications
摘要：
只有实根的一元多项式--虽然很特殊--确实经常出现，它们的性质可以在不同的领域得出有趣的结论。由于最近的工作主要是两个年轻的数学家，朱利叶斯Borcea和彼得Brändén，一个非常成功的多元推广这种方法已经制定。本文的第一部分调查的“多元稳定”的多项式理论的一些主要成果-这些结果的最核心的是线性变换保持稳定的多项式的表征。第二部分介绍了这一理论在复分析、矩阵论、概率论和统计力学以及组合数学中的各种应用。
英文摘要：
Univariate polynomials with only real roots -- while special -- do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Brändén, a very successful multivariate generalization of this method has been developed. The first part of this paper surveys some of the main results of this theory of "multivariate stable" polynomials -- the most central of these results is the characterization of linear transformations preserving stability of polynomials. The second part presents various applications of this theory in complex analysis, matrix theory, probability and statistical mechanics, and combinatorics.
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