Algebra/Topology Seminar

Speaker: Jurij Volcic

Title: Determinantal zeros of noncommutative polynomials

Abstract: Hilbert's Nullstellensatz is one of the most fundamental correspondences between algebra and geometry. In recent years, there has been an emerging interest in polynomial equations and inequalities in several matrix variables, prompted by developments in control theory, quantum information, matrix analysis and polynomial optimization. The arising problems call for a suitable version of (real) algebraic geometry in matrix variables.

This talk concerns a Nullstellensatz fitting this context. Given a noncommutative polynomial f (an element of a free algebra), its free locus Z(f) is the collection of matrix tuples X such that f(X) is a singular matrix. The main results of the talk are the correspondence between components of the Z(f) and irreducible factors of f, and a Nullstellensatz for free loci: a noncommutative polynomial g attains singular values wherever f attains singular values if and only if irreducible factors of f are essentially factors of g. An analog of Bertini's irreducibility theorem for the free algebra will also be given. Finally, applications for noncommutative polynomial inequalities will be discussed.