## Abstract

This paper solves the rational noncommutative analogue 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 generalisation 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 ≥ 0 if and only if it belongs to the rational quadratic module generated by L. The essential intermediate step toward this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils.

Original language | English |
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Article number | e61 |

Journal | Forum of Mathematics, Sigma |

Volume | 9 |

DOIs | |

State | Published - 1 Jan 2021 |

Externally published | Yes |

## Keywords

- free skew field
- Hilbert’s 17th problem
- linear matrix inequality
- linear matrix pencil
- Noncommutative rational function
- Positivstellensatz
- spectrahedron

## ASJC Scopus subject areas

- Analysis
- Theoretical Computer Science
- Algebra and Number Theory
- Statistics and Probability
- Mathematical Physics
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Mathematics