A note on the nonexistence of sum of squares certificates for the Bessis–Moussa–Villani conjecture

Abstract

The algebraic reformulation of the Bessis–Moussa–Villani conjecture is equivalent to a family of dimension-free tracial inequalities involving positive semidefinite matrices. Sufficient conditions for these to hold in the form of algebraic identities involving polynomials in noncommuting variables have been given by Klep and Schweighofer [“Sums of Hermitian squares and the BMV conjecture,” J. Stat. Phys. 133, 739 (2008)]. Later the existence of these certificates has been settled for all but one case, which is resolved in this note.