Honors Thesis Presentation: "Commutativity of Generic Solutions to Polynomials in Matrices over Finite Fields"

Speaker: Sam Heil, Washington University in Saint Louis

Abstract: Recent developments in analysis have examined the class of nc polynomials, polynomials in non-commuting variables often evaluated over square matrices. Agler and McCarthy proved in 2014 that all solutions Y to a generic two-variable polynomial p(X, Y) for a fixed n by n complex matrix must commute with X, for certain genericity conditions satisfied by almost all matrices and two-variable polynomials. In this paper, we extend these results to matrices over finite fields for a limited class of two-variable free polynomials. For polynomials of the form f(x, y) = a_0y^n+a_1y^{n-1}x+...+a_nx^n, we construct a sufficient condition on polynomial-matrix pairs (f, X) implying that any solution Y to f(X, Y) = 0 commutes with X. Considering matrices over F_q asymptotically as q approaches infinity, we show that for each polynomial f of the form above, all but O(1/q) matrices X satisfy the condition, thus yielding only commuting solution pairs.

Host: John Shareshian

(Access Zoom Presentation HERE)