Geometry of free loci and factorization of noncommutative polynomials

TL;DR

This paper explores the geometric structure of noncommutative polynomials' singularity loci, linking irreducibility of the polynomial to the geometric irreducibility of associated hypersurfaces, with applications in algebra, perturbation theory, and geometry.

Contribution

It establishes a fundamental connection between polynomial irreducibility and the geometric irreducibility of free loci, introducing new irreducibility results for linear pencils.

Findings

Irreducibility of f iff Z_n(f) is eventually irreducible

New irreducibility results for linear pencils

Applications to invariant subspaces and linear matrix inequalities

Abstract

The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.