Real Algebraic and Analytic Geometry
Submission: 2008, August 11.
We show that all the coefficients of the polynomial $$\tr((A+tB)^m)\in\R[t]$$ are nonnegative whenever $m\le 13$ is a nonnegative integer and $A$ and $B$ are positive semidefinite matrices of the same size. This has previously been known only for $m\le 7$. The validity of the statement for arbitrary $m$ has recently been shown to be equivalent to the Bessis-Moussa-Villani conjecture from theoretical physics. 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.
Mathematics Subject Classification (2000): 11E25, 13J30, 15A90, 15A45, 08B20, 90C22.
Keywords and Phrases: Bessis-Moussa-Villani (BMV) conjecture, sum of hermitian squares, trace inequality, semidefinite programming.
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