Sums of hermitian squares and the BMV conjecture

Abstract

Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. This has previously been known only up to length 7. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.

21 pages; minor changes; a companion Mathematica notebook is now available in the source file

References in corpus (6)

• Connes' embedding conjecture and sums of hermitian squares
• Positive polynomials in scalar and matrix variables, the spectral theorem and optimization
• Proof of the cases $p\le 7$ of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture
• Asymptotic Positivity of Hurwitz Product Traces
• Remarks on BMV conjecture
• Sums of Hermitian Squares as an Approach to the BMV Conjecture

Cited by in corpus (11)

• Sparse Noncommutative Polynomial Optimization
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• Sum--of--squares results for polynomials related to the Bessis--Moussa--Villani conjecture
• Semidefinite programming in matrix unknowns which are dimension free
• The singular bivariate quartic tracial moment problem
• On D. Haegele's approach to the Bessis-Moussa-Villani conjecture
• Proof of the BMV Conjecture
• Sums of Hermitian Squares as an Approach to the BMV Conjecture
• Stahl's Theorem (aka BMV Conjecture): Insights and Intuition on its Proof
• Equations solvable by radicals in a uniquely divisible group
• On a polynomial positivity question of Collins, Dykema, and Torres-Ayala related to the Bessis-Moussa-Villani Conjecture