Hilbert's 17th problem in free skew fields

TL;DR

This paper proves that noncommutative rational functions positive on all hermitian matrix tuples are sums of hermitian squares, extending positivity certificates and establishing a Positivstellensatz for free spectrahedra.

Contribution

It generalizes positivity certificates to noncommutative rational functions with singularities and provides a Positivstellensatz for free spectrahedra.

Findings

Noncommutative rational functions positive on all hermitian matrices are sums of hermitian squares.

Established a Positivstellensatz for free spectrahedra involving rational functions.

Extended theorem for invertible evaluations of linear matrix pencils.

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.