Sums of Hermitian Squares and the BMV Conjecture
By Igor Klep, Markus Schweighofer. 2007
57 citations recorded by the source corpus.
We show that all the coefficients of the polynomial $$\mathop{\mathrm{tr}}((A+tB)^{m})\in\mathbb{R}[t]$$ are nonnegative whenever m≤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≤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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/27ee5ff20ebbc80c6d0f1ec31438eadec87f41aa
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In Algebra and Number Theory
In Computational Theory and Mathematics
In Numerical Analysis
Citations in the corpus
- Proof of the BMV conjecture (2011, background): “Recent publications are typically concerned with techniques from non-communitative algebra and combinatorics ([8], [10], [6], [9], [7], [11], [12], [2], [5]).”