Determinantal zeros and factorization of noncommutative polynomials

Algebra Seminar
Monday, February 12, 2024 - 1:00pm for 1 hour (actually 50 minutes)
Skiles 005
Jurij Volčič – Drexel University
Changxin Ding

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30am in Skiles 005.

Hilbert's Nullstellensatz about zero sets of polynomials is one of the most fundamental correspondences between algebra and geometry. More recently, there has been an emerging interest in polynomial equations and inequalities in several matrix variables, prompted by developments in control systems, quantum information theory, operator algebras and optimization. The arising problems call for a suitable version of (real) algebraic geometry in noncommuting variables; with this in mind, the talk considers matricial sets where noncommutative polynomials attain singular values, and their algebraic counterparts.

Given a polynomial f in noncommuting variables, its free (singularity) locus is the set of all matrix tuples X such that f(X) is singular.  The talk focuses on the interplay between geometry of free loci (irreducible components, inclusions, eigenlevel sets, smooth points) and factorization in the free algebra. In particular, a Nullstellensatz for free loci is given, as well as a noncommutative variant of Bertini's irreducibility theorem and its consequences.