Sums of Hermitian Squares and the BMV Conjecture
نویسندگان
چکیده
منابع مشابه
Sums of Hermitian Squares and the Bmv Conjecture
We show that all the coefficients of the polynomial tr((A+ tB)) ∈ 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 o...
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Lieb and Seiringer stated in their reformulation of the BessisMoussa-Villani conjecture that all coefficients of the polynomial p(t) = tr[(A+ B)] are nonnegative whenever A and B are any two positive semidefinite matrices of the same size. We will show that for all m ∈ N the coefficient of t in p(t) is nonnegative, using a connection to sums of Hermitian squares of non-commutative polynomials w...
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We show that Connes’ embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is nonnegative whenever self-adjoint contraction matrices of the same size are substituted for the variables. These algebraic certificates involve sums of hermitia...
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This paper presents an algorithm and its implementation in the software package NCSOStools for finding sums of hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allows us to use semidefinite programming. For rational polynomials num...
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We show that Connes’ embedding conjecture (CEC) is equivalent to a real version of the same (RCEC). Moreover, we show that RCEC is equivalent to a real, purely algebraic statement concerning trace positive polynomials. This purely algebraic reformulation of CEC had previously been given in both a real and a complex version in a paper of the last two authors. The second author discovered a gap i...
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ژورنال
عنوان ژورنال: Journal of Statistical Physics
سال: 2008
ISSN: 0022-4715,1572-9613
DOI: 10.1007/s10955-008-9632-x