Slices of Stable Polynomials and Connections to the Grace-Walsh-Szeg\H{o} theorem

Abstract

Univariate polynomials are called stable with respect to a domain $D$ if all of their roots lie in $D$. We study linear slices of the space of stable univariate polynomials with respect to a half-plane. We show that a linear slice always contains a stable polynomial with only a few distinct roots. Subsequently, we apply these results to symmetric polynomials and varieties. We show that for varieties defined by few multiaffine symmetric polynomials, the existence of a point in $D^n$ with few distinct coordinates is necessary and sufficient for the intersection with $D^n$ to be non-empty. This is at the same time a generalization of the so-called degree principle to stable polynomials and a result similar to Grace-Walsh-Szeg\H{o}'s coincidence theorem.