A converse to the Grace--Walsh--Szeg\H{o} theorem

TL;DR

This paper characterizes when symmetrizers preserve polynomial stability, showing that permutation invariance alone is insufficient, and introduces a new algebraic structure for Grace-like polynomials.

Contribution

It provides a necessary and sufficient condition for symmetrizers to preserve stability, and introduces a natural multiplication for Grace-like polynomials.

Findings

Symmetrizer preserves stability iff the group is orbit homogeneous.

Permutation invariance alone does not ensure stability preservation.

Grace-like polynomials form an algebra under a new multiplication.

Abstract

We prove that the symmetrizer of a permutation group preserves stability of a polynomial if and only if the group is orbit homogeneous. A consequence is that the hypothesis of permutation invariance in the Grace-Walsh-Szeg\H{o} Coincidence Theorem cannot be relaxed. In the process we obtain a new characterization of the \emph{Grace-like polynomials} introduced by D. Ruelle, and prove that the class of such polynomials can be endowed with a natural multiplication.