Undergraduate Seminar: "Commutativity of Generic Solutions to Polynomials in Matrices over Finite Fields"
Abstract: Agler and McCarthy proved in 2014 that all solutions Y to a generic two-variable polynomial p(X, Y) for a fixed generic n x n matrix X with complex entries must commute with X, for certain genericity conditions satisfied by almost all matrices and two-variable polynomials. 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_0x^n + a_1x^(n-1)y + ... + a_ny^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 the finite field 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: Adeli Hutton