Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions

A real univariate polynomial of degree n is called hyperbolic if all of its n roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article is on families of hyperbolic polynomials which are determined through k linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a hyperbolic slice. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the local extreme points of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most k distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example, to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

Automated summary

This article focuses on the geometry of a class of hyperbolic polynomial families determined by linear conditions on the coefficients. These polynomials have all their roots on the real line. The set of hyperbolic polynomials is stratified according to the multiplicities of the real zeros, and this stratification also applies to the hyperbolic slices. The study shows that the local extreme points of hyperbolic slices correspond to hyperbolic polynomials with at most k distinct roots, and that the convex hull of such a family is generally a polyhedron. The article also explores the implications of these results for symmetric real varieties and symmetric semi-algebraic sets, particularly in terms of sparse representations and sampling.