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“Positive” noncommutative polynomials are sums of squares. (English) Zbl 1033.12001

It is well known that any positive polynomial is always a sum of squares of rational functions but not always a sum of squares of polynomials.
The paper studies Hilbert’s 17th problem for symmetric noncommutative polynomials.
The main result is, “A symmetric noncommutative polynomial is a sum of squares of polynomials if and only if it is matrix positive,” where matrix positive means that whenever matrices of any size are substituted for variables in the polynomial, the matrix value which the polynomial takes is positive semidefinite.
But in fact this is of no use to check whether or not a polynomial is a sum of squares of polynomials because it is impractical. The result can rather be used in the converse way: use algebraic and algorithmic methods from Reznick, Powers-Wörmann and Parrilo to check if the polynomial is a sum of squares of polynomials, and then deduce that the polynomial is or is not matrix positive.

MSC:

12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
13J30 Real algebra
14P99 Real algebraic and real-analytic geometry
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