Matrix Fej\'er-Riesz theorem with gaps

TL;DR

This paper extends the matrix Fejér-Riesz theorem to arbitrary closed semialgebraic sets, providing new characterizations for positive semidefinite matrix polynomials, especially highlighting differences between compact and non-compact cases.

Contribution

It generalizes the matrix Fejér-Riesz theorem to broader sets using real algebraic geometry, including algebraic curves, and discusses conditions with and without denominators.

Findings

Denominator-free characterization exists for compact sets.

Counterexamples show limitations in the non-compact case.

Weaker characterizations with denominators are available in non-compact cases.

Abstract

The matrix Fej\'er-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line $\mathbb{R}$. We extend a characterization to arbitrary closed semialgebraic sets $K\subseteq \mathbb{R}$ by the use of matrix preorderings from real algebraic geometry. In the compact case a denominator-free characterization exists, while in the non-compact case there are counterexamples. However, there is a weaker characterization with denominators in the non-compact case. At the end we extend the results to algebraic curves.