Sums of Hermitian Squares and the BMV Conjecture

作     者：Klep, Igor Schweighofer, Markus 

作者机构：Univ Ljubljani Oddelek Matemat Inst Matemat Ljubljana 1111 Slovenia Univ Rennes 1 Math Lab F-35042 Rennes France 

出 版 物：《JOURNAL OF STATISTICAL PHYSICS》 (统计物理学杂志)

年 卷 期：2008年第133卷第4期

页      面：739-760页

核心收录：

学科分类：07[理学] 070201[理学-理论物理] 0702[理学-物理学] 

基　　金：Deutsche Forschungsgemeinschaft, DFG Universität Konstanz Javna Agencija za Raziskovalno Dejavnost RS, ARRS, (Z1-9570-0101-06) 

主　　题：Bessis-Moussa-Villani (BMV) conjecture Sum of hermitian squares Trace inequality Semidefinite programming 

摘      要：We show that all the coefficients of the polynomial are nonnegative whenever = 13 is a nonnegative integer and A and B are positive semidefinite matrices of the same size. This has previously been known only for = 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.