Abstract
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 $L\succeq 0$ 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.
Highlights
In his famous problem list of 1900, Hilbert asked whether every positive rational function can be written as a sum of squares of rational functions
Thisfree real algebraic geometry started with the seminal work of Helton [Hel02] and McCullough [McC01], who proved that a noncommutative polynomial is positive semidefinite on all tuples of Hermitian matrices precisely when it is a sum of Hermitian squares of noncommutative polynomials
The purpose of this paper is to extend this result to noncommutative rational functions
Summary
In his famous problem list of 1900, Hilbert asked whether every positive rational function can be written as a sum of squares of rational functions. The second direction attempts to answer questions about the positivity of noncommutative polynomials and rational functions when matrix arguments of all finite sizes are considered. Such questions naturally arise in control systems [dOHMP09], operator algebras [Oza16] and quantum information theory [DLTW08, P-KRR+19]. The solution of Hilbert’s 17th problem in the free skew field presented in this paper (Corollary 5.4) states that every r ∈ C) , positive semidefinite on its Hermitian domain, is a sum of Hermitian squares in C) This statement was proved in [KPV17] for noncommutative rational functions r that are regular, meaning that r( ) is well-defined for every tuple of Hermitian matrices. A Nullstellensatz for cancellation of nonHermitian singularities is given in Proposition 6.3
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