"Positive" Noncommutative Polynomials Are Sums of Squares

Abstract

Hilbert's 17th problem concerns expression of polynomials on R n as a sum of squares. It is well known that many positive polynomials are not sums of squares; see [Re], [D'A] for excellent surveys. In this paper we consider symmetric noncommutative 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 semidefinite. The result in this paper is: A polynomial is matrix-positive if and only if it is a sum of squares.