\Positive" Non-Commutative Polynomials are Sums of Squares
نویسنده
چکیده
Hilbert's 17th problem concerns expressing polynomials on R as a sum of squares. It is well known that many positive polynomials are not sums of squares; see [R00] [deA preprt] for excellent surveys. In this paper we consider symmetric non-commutative polynomials and call one \matrix positive", if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semide nite. The result in this paper is: A polynomial is matrix positive if and only if it is a sum of squares.
منابع مشابه
Symmetric, Positive Semidefinite Polynomials Which
This paper presents a construction for symmetric, non-negative polynomials, which are not sums of squares. It explicitly generalizes the Motzkin polynomial and the Robinson polynomials to families of non-negative polynomials, which are not sums of squares. The degrees of the resulting polynomials can be chosen in advance. 2000 Mathematics Subject Classification: 12Y05, 20C30, 12D10, 26C10, 12E10
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