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An elementary and constructive solution to Hilbert’s 17th problem for matrices. (English) Zbl 1126.12001

A real matrix A is called positive semidefinite if it is symmetric with all nonnegative eigenvalues. The main aim of the paper is the following theorem: If a symmetric matrix A with entries in R[x 1 ,,x n ] is positive semidefinite for all sustitutions (x 1 ,,x n ) n , then A can be expressed as a sum of squares of symmetric matrices with entries in R(x 1 ,,x n )·

This generalizes the famous Artin’s theorem solving Hilbert’s 17th problem. The theorem proved by the authors was originally proved by C. Procesi and M. Schacher [Ann. Math. (2) 104, 395–406 (1976; Zbl 0347.16010)]. However, the proof in the paper under review is constructive modulo the fact that one can represent the entries of A as a sum of rational functions.

MSC:
12D15Formally real fields
13J30Real algebra
15A21Canonical forms, reductions, classification
15A54Matrices over function rings