A real matrix 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 with entries in is positive semidefinite for all sustitutions , then can be expressed as a sum of squares of symmetric matrices with entries in
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 as a sum of rational functions.