Geometry of free loci and factorization of noncommutative polynomials

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.

v2: 32 pages, includes a table of contents

References in corpus (7)

• The Lax conjecture is true
• Dilations, inclusions of matrix convex sets, and completely positive maps
• Extreme points of matrix convex sets, free spectrahedra and dilation theory
• Free loci of matrix pencils and domains of noncommutative rational functions
• Bianalytic Maps Between Free Spectrahedra
• On the Factorization of Non-Commutative Polynomials (in Free Associative Algebras)
• Quadratic Modules, C*-Algebras, and Free Convexity

Cited by in corpus (9)

• Determinantal hypersurfaces and representations of Coxeter groups
• Bianalytic free maps between spectrahedra and spectraballs
• Noncommutative polynomials describing convex sets
• Ranks of linear matrix pencils separate simultaneous similarity orbits
• A Factorization Theory for some Free Fields
• Factorization of noncommutative polynomials and Nullstellensätze for the free algebra
• A Standard Form in (some) Free Fields: How to construct Minimal Linear Representations
• Matrix Extreme Points and Free extreme points of Free spectrahedra
• Spectral surfaces for operator pairs and Hadamard matrices of F type