Polynomial equations for matrices over integers modulo a prime power and the cokernel of a random matrix

Abstract

Given a prime p and a positive integer k, let Mn(Z/pkZ) be the ring of n×n matrices over Z/pkZ. We consider the number of solutions X∈Mn(Z/pkZ) to the polynomial equation P(X)=0, where P(t) is a monic polynomial in (Z/pkZ)[t] whose reduction modulo p is square-free over the finite field Fp of p elements. Noting that P(X)=0 if and only if cok(P(X))≃(Z/pkZ)n, we give a conjectural generalization of counting solutions to P(X)=0 as the distribution of the cokernel cok(P(X)) of P(X) up to isomorphisms, where X is a uniform random matrix in Mn(Z/pkZ). This distribution involves an explicit formula when we fix the residue class of X modulo p. We prove this conjecture for the special case when the image of P(t) in Fp[t] modulo p is irreducible. We explain how the distribution we obtain is closely related to the Cohen-Lenstra distribution in number theory. Our proof involves algebraic and combinatorial arguments in linear algebra over Z/pkZ and builds upon a previous work of Cheong and Kaplan.