Positive univariate trace polynomials

TL;DR

This paper studies the positivity of univariate trace polynomials on symmetric matrices, providing a tracial analog of Artin's solution to Hilbert's 17th problem, characterizing positive semidefinite cases.

Contribution

It introduces a tracial version of Artin's theorem, characterizing positive univariate trace polynomials as quotients of sums of squares and traces of squares.

Findings

Positive semidefinite trace polynomials are quotients of sums of squares and traces of squares.

The paper extends classical positivity results to the setting of trace polynomials.

Provides a framework for understanding positivity in matrix evaluations of trace polynomials.

Abstract

A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(x^j). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.