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"Positive" Noncommutative Polynomials Are Sums of Squares

J. William Helton
Annals of Mathematics
Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 675-694
Published by: Annals of Mathematics
DOI: 10.2307/3597203
Stable URL: http://www.jstor.org/stable/3597203
Page Count: 20
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"Positive" Noncommutative Polynomials Are Sums of Squares
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Abstract

Hilbert's 17th problem concerns expression of polynomials on Rn 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.

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