Slices of Stable Polynomials and Connections to the Grace-Walsh-Szegő theorem

Abstract: Univariate polynomials are called stable with respect to a circular region $\mathcal{A}$, if all of their roots are in $\mathcal{A}$. We consider the special case where $\mathcal{A}$ is a half-plane and investigate affine slices of the set of stable polynomials. In this setup, we show that an affine slice of codimension $k$ always contains a stable polynomial that possesses at most $2(k+2)$ distinct roots on the boundary and at most $(k+2)$ distinct roots in the interior of $\mathcal{A}$. This result also extends to affine slices of weakly Hurwitz polynomials. Subsequently, we apply these results to symmetric polynomials and varieties. Here we show that it is necessary and sufficient for a variety described by polynomials in few multiaffine polynomials to contain points in $\mathcal{A}^n$ with few distinct coordinates for its intersection with $\mathcal{A}^n$ being non-empty. This is at the same time a generalization of the degree principle to stable polynomials and a result similar to Grace-Walsh-Szeg\H{o}'s coincidence theorem on multiaffine symmetric polynomials.