Free Bertini's theorem and applications.

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 ƒ is a noncommutative polynomial such that ƒ-λ factors for infinitely many scalars λ, then there exist a noncommutative polynomial h and a nonconstant univariate polynomial p such that ƒ = p ○ h. Two applications of free Bertini's theorem for matrix evaluations of noncommutative polynomials are given. An eigenlevel set of ƒ is the set of all matrix tuples X where ƒ(X) attains some given eigenvalue. It is shown that eigenlevel sets of ƒ and g coincide if and only if ƒa = ag for some nonzero noncommutative polynomial a. The second application pertains to quasiconvexity and describes polynomials ƒ such that the connected component of X tuple of symmetric n × n matrices: λI ≻ ƒ(X) about the origin is convex for all natural n and λ > 0. It is shown that such a polynomial is either everywhere negative semidefinite or the composition of a univariate and a convex quadratic polynomial. [ABSTRACT FROM AUTHOR]