Free Bertini's theorem and applications

TL;DR

This paper extends Bertini's theorem to free algebras, characterizing noncommutative polynomials with infinitely many factorings and exploring applications in matrix eigenvalues and convexity.

Contribution

It introduces a free algebra analog of Bertini's theorem, providing new insights into noncommutative polynomial factorizations and their applications.

Findings

Noncommutative polynomial factoring characterized by composition with univariate polynomials.

Eigenlevel sets of polynomials coincide iff they differ by a noncommutative polynomial factor.

Polynomials with convex matrix evaluations are either negative semidefinite or composed of convex quadratic and univariate polynomials.

Abstract

The simplest version of Bertini's irreducibility theorem states that the generic fiber of a non-composite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if $f$ is a noncommutative polynomial such that $f-\lambda$ factors for infinitely many scalars $\lambda$, then there exist a noncommutative polynomial $h$ and a nonconstant univariate polynomial $p$ such that $f=p\circ h$. Two applications of free Bertini's theorem for matrix evaluations of noncommutative polynomials are given. An eigenlevel set of $f$ is the set of all matrix tuples $X$ where $f(X)$ attains some given eigenvalue. It is shown that eigenlevel sets of $f$ and $g$ coincide if and only if $fa=ag$ for some nonzero noncommutative polynomial $a$. The second application pertains quasiconvexity and describes polynomials $f$ such that the connected component of $\{X \text{ tuple of symmetric $n\times n$ matrices}: \lambda I\succ f(X) \}$ about the origin is convex for all natural $n$ and $\lambda>0$. It is shown that such a polynomial is either everywhere negative semidefinite or the composition of a univariate and a convex quadratic polynomial.