论文标题
组合和保存圆锥形稳定的多项式
Combinatorics and preservation of conically stable polynomials
论文作者
论文摘要
给定一个封闭的，凸锥$ k \ subseteq \ mathbb {r}^n $，是\ mathbb {c} [\ mathbf {z} $的多变量多项式$ f \，如果其根部不包含$ k $ k $ k $ k $ k $ k $。如果$ k $是非负晶体，则$ k $稳定性专门针对多项式稳定性的通常概念。 我们从通常的稳定性方面的保存操作和组合标准的概括。一个特殊的重点是阳性半足质矩阵（PSD稳定性）的锥。特别是，我们证明了在反转操作员的自然概括下保存PSD稳定性。此外，我们提供有关PSD稳定多项式支持的条件，并表征了PSD稳定多项式特殊系列的支持。
Given a closed, convex cone $K\subseteq \mathbb{R}^n$, a multivariate polynomial $f\in\mathbb{C}[\mathbf{z}]$ is called $K$-stable if the imaginary parts of its roots are not contained in the relative interior of $K$. If $K$ is the non-negative orthant, $K$-stability specializes to the usual notion of stability of polynomials. We develop generalizations of preservation operations and of combinatorial criteria from usual stability towards conic stability. A particular focus is on the cone of positive semidefinite matrices (psd-stability). In particular, we prove the preservation of psd-stability under a natural generalization of the inversion operator. Moreover, we give conditions on the support of psd-stable polynomials and characterize the support of special families of psd-stable polynomials.
