Sum--of--squares results for polynomials related to the Bessis--Moussa--Villani conjecture

TL;DR

This paper investigates sum-of-squares representations of specific polynomials related to the Bessis--Moussa--Villani conjecture, showing limitations for certain cases and confirming nonnegativity in others, impacting approaches to the conjecture.

Contribution

It proves that certain polynomials cannot be expressed as sums of commutators and Hermitian squares for many cases, narrowing the scope of sum-of-squares methods for the BMV conjecture.

Findings

Polynomials S_{m,k}(A,B) are not sums of commutators and Hermitian squares for even m,k with 6 <= k <= m-10.

S_{m,4}(A,B) equals a sum of commutators and Hermitian squares when m is even and not divisible by 4.

Trace of S_{m,4}(A,B) is nonnegative for all Hermitian matrices A and B in specified cases.

Abstract

We show that the polynomial S_{m,k}(A,B), that is the sum of all words in noncommuting variables A and B having length m and exactly k letters equal to B, is not equal to a sum of commutators and Hermitian squares in the algebra R<X,Y> where X^2=A and Y^2=B, for all even values of m and k with 6 <= k <= m-10, and also for (m,k)=(12,6). This leaves only the case (m,k)=(16,8) open. This topic is of interest in connection with the Lieb--Seiringer formulation of the Bessis--Moussa--Villani conjecture, which asks whether the trace of S_{m,k}(A,B)) is nonnegative for all positive semidefinite matrices A and B. These results eliminate the possibility of using "descent + sum-of-squares" to prove the BMV conjecture. We also show that S_{m,4}(A,B) is equal to a sum of commutators and Hermitian squares in R<A,B> when m is even and not a multiple of 4, which implies that the trace of S_{m,4}(A,B) is nonnegative for all Hermitian matrices A and B, for these values of m.