Null- and Positivstellens\"atze for rationally resolvable ideals

TL;DR

This paper extends Nullstellensatz concepts to free algebras with rational relations, characterizing polynomial vanishing conditions on noncommutative structures like spherical isometries and unitaries, with implications for free real algebraic geometry.

Contribution

It introduces Nullstellensatz results for rationally resolvable ideals in free algebras, linking algebraic properties to embeddability into skew fields and extending to involution cases.

Findings

Nullstellensatz holds for rationally resolvable ideals in free algebras.

Polynomials vanishing on spherical isometries belong to specific two-sided ideals.

Positivity on spherical isometries implies sums of Hermitian squares modulo the ideal.

Abstract

Hilbert's Nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in C[x]. In the free algebra C<X> the vanishing set of a two-sided ideal I is defined in a dimension-free way using images in finite-dimensional representations of C<x>/I. In this article Nullstellens\"atze for a simple but important class of ideals in the free algebra - called tentatively rationally resolvable here - are presented. An ideal is rationally resolvable if its defining relations can be eliminated by expressing some of the X variables using noncommutative rational functions in the remaining variables. Whether such an ideal I satisfies the Nullstellensatz is intimately related to embeddability of C<x>/I into (free) skew fields. These notions are also extended to free algebras with involution. For instance, it is proved that a polynomial vanishes on all tuples of spherical isometries iff it is a member of the two-sided ideal I generated by 1-\sum_j X_j^* X_j. This is then applied to free real algebraic geometry: polynomials positive semidefinite on spherical isometries are sums of Hermitian squares modulo I. Similar results are obtained for nc unitary groups.