Free Bertini’s theorem and applications

Abstract

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