Matrix versions of real and quaternionic nullstellensatz

TL;DR

This paper extends the Nullstellensatz to matrix polynomials over real and quaternionic fields, simplifying definitions and broadening applicability in non-commutative algebraic geometry.

Contribution

It introduces matrix versions of the Nullstellensatz for real and quaternionic polynomials, improving the real case by simplifying the real left ideal definition.

Findings

Extended Nullstellensatz to matrix polynomials over real and quaternionic fields.

Simplified the definition of real left ideals in matrix polynomial context.

Provided methods for non-commutative and real algebraic geometry extensions.

Abstract

Real Nullstellensatz is a classical result from Real Algebraic Geometry. It has recently been extended to quaternionic polynomials by Alon and Paran. The aim of this paper is to extend their Quaternionic Nullstellensatz to matrix polynomials. We also obtain an improvement of the Real Nullstellensatz for matrix polynomials in the sense that we simplify the definition of a real left ideal. We use the methods from the proof of the matrix version of Hilbert's Nullstellensatz and we obtain their extensions to a mildly non-commutative case and to the real case.