Total nonnegativity and stable polynomials

TL;DR

This paper establishes a deep connection between the stability of certain multiaffine polynomials and total nonnegativity in the Grassmannian, using matrix actions and classical positivity theories.

Contribution

It characterizes stable multiaffine polynomials with Plücker coordinates as exactly those associated with totally nonnegative Grassmannian points, linking stability to total nonnegativity.

Findings

Polynomial stability iff Grassmannian point is totally nonnegative

Matrix actions preserve stability iff matrices are totally nonnegative

Uses classical total nonnegativity and Pólya-Schur theories

Abstract

We consider homogeneous multiaffine polynomials whose coefficients are the Pl\"ucker coordinates of a point $V$ of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if $V$ is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix $A$ preserves stability of polynomials if and only if $A$ is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized P\'olya-Schur theory of Borcea and Br\"and\'en.