Stable Polynomials via Undirected Colored Graphs

TL;DR

This paper explores the relationship between undirected colored graphs and stable polynomials, analyzing their boundary zeros and contact order using linear algebra techniques to deepen understanding of their properties.

Contribution

It introduces a systematic study linking colored graphs to stable polynomials and characterizes boundary zeros and contact order, advancing theoretical understanding.

Findings

Characterization of boundary zeros of stable polynomials

Analysis of contact order near boundary zeros

Use of linear algebra techniques for polynomial properties

Abstract

This paper initiates a systematic study of connections between undirected colored graphs and associated two-variable stable polynomials obtained via Cauchy transform-type formulas. Examples of such stable polynomials have played crucial roles in other recent studies, though their general properties have remained rather opaque. Using linear algebra techniques, this paper characterizes when these stable polynomials have boundary zeros and studies the finer behavior, called contact order, of their zero sets near the guaranteed boundary zeros.