Finsler's Lemma for Matrix Polynomials

TL;DR

This paper extends Finsler's Lemma to symmetric matrix polynomials with an additional negativity condition, providing new characterizations and applications in Noncommutative Real Algebraic Geometry.

Contribution

It generalizes Finsler's Lemma to matrix polynomials under a negativity assumption, linking classical results to noncommutative algebraic geometry.

Findings

Extended Finsler's Lemma to matrix polynomials with negativity condition

Provided applications to Noncommutative Real Algebraic Geometry

Reduced to classical characterizations for the case n=1

Abstract

Finsler's Lemma charactrizes all pairs of symmetric $n \times n$ real matrices $A$ and $B$ which satisfy the property that $v^T A v>0$ for every nonzero $v \in \mathbb{R}^n$ such that $v^T B v=0$. We extend this characterization to all symmetric matrices of real multivariate polynomials, but we need an additional assumption that $B$ is negative semidefinite outside some ball. We also give two applications of this result to Noncommutative Real Algebraic Geometry which for $n=1$ reduce to the usual characterizations of positive polynomials on varieties and on compact sets.