Further techniques on a polynomial positivity question of Collins, Dykema, and Torres-Ayala

TL;DR

This paper proves that the coefficient of t^2 in the trace of (A+tB)^6 is a sum of squares in the entries of symmetric matrices A and B, contributing to polynomial positivity questions.

Contribution

It establishes that a specific polynomial coefficient related to symmetric matrices can be expressed as a sum of squares, advancing understanding in polynomial positivity.

Findings

Coefficient of t^2 is a sum of squares in matrix entries

Provides a new sum of squares representation for polynomial coefficients

Advances polynomial positivity theory in matrix analysis

Abstract

We prove that the coefficient of $t^2$ in $\mathsf{trace}((A+tB)^6)$ is a sum of squares in the entries of the symmetric matrices $A$ and $B$.