Hilbert's 17th problem in free skew fields

Abstract

This paper solves the rational noncommutative analog of Hilbert's 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of hermitian matrices in its domain, then it is a sum of hermitian squares of noncommutative rational functions. This result is a generalization and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every matricial solution of a linear matrix inequality $L\succeq0$ if and only if it belongs to the rational quadratic module generated by $L$. The essential intermediate step towards this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils.