A Nichtnegativstellensatz for Polynomials in Noncommuting Variables
نویسنده
چکیده
Let S ∪ {f} be a set of symmetric polynomials in noncommuting variables. If f satisfies a polynomial identity P i h ∗ i fhi = 1 + P i g ∗ i sigi for some si ∈ S ∪ {1}, then f is obviously nowhere negative semidefinite on the class of tuples of non-zero operators defined by the system of inequalities s ≥ 0 (s ∈ S). We prove the converse under the additional assumption that the quadratic module generated by S is archimedean.
منابع مشابه
NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials
NCSOStools is a Matlab toolbox for • symbolic computation with polynomials in noncommuting variables; • constructing and solving sum of hermitian squares (with commutators) programs for polynomials in noncommuting variables. It can be used in combination with semidefinite programming software, such as SeDuMi, SDPA or SDPT3 to solve these constructed programs. This paper provides an overview of ...
متن کاملTensor Algebras and Displacement Structure. Ii. Noncommutative Szeg¨o Polynomials
In this paper we continue to explore the connection between tensor algebras and displacement structure. We focus on recursive orthonormalization and we develop an analogue of the Szegö type theory of orthogonal polynomials in the unit circle for several noncommuting variables. Thus, we obtain the recurrence equations and Christoffel-Darboux formulas for Szegö polynomials in several noncommuting...
متن کاملSemidefinite Programming and Sums of Hermitian Squares of Noncommutative Polynomials
An algorithm for finding sums of hermitian squares decompositions for polynomials in noncommuting variables is presented. The algorithm is based on the “Newton chip method”, a noncommutative analog of the classical Newton polytope method, and semidefinite programming.
متن کاملClassification of All Noncommutative Polynomials Whose Hessian Has Negative Signature One and a Noncommutative Second Fundamental Form
Every symmetric polynomial p = p(x) = p(x1, . . . , xg) (with real coefficients) in g noncommuting variables x1, . . . , xg can be written as a sum and difference of squares of noncommutative polynomials:
متن کاملGeneralizing Rules of Three
This behavior has been seen in some notable cases. Kirillov [3] shows that elementary symmetric polynomials in noncommuting variables commute (and, in some cases, all Schur functions) when elementary symmetric polynomials of degree at most three commute when restricted to at most three of the variables. Generalizing this, Blasiak and Fomin [1] give a wider theory for rules of three of generatin...
متن کامل