Colloquium: Tuesday, March 8, 2022. Speaker: Jurij Volcic (University of Copenhagen). Title: “Determinantal zeros of noncommutative polynomials”.
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Abstract: Hilbert’s Nullstellensatz on zero sets of polynomials 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, operator algebras and polynomial optimization.
The arising problems call for a suitable version of (real) algebraic geometry in noncommuting variables.
This talk concerns a Nullstellensatz fitting this context.
Given a noncommutative polynomial f (an element of a free algebra), its free locus is the collection of matrix tuples X such that f(X) is a singular matrix. This is a noncommutative analog of a hypersurface in algebraic geometry. The main results of the talk are the correspondence between components of the free locus of 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. Several applications for noncommutative polynomial inequalities will also be discussed.