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Given a prime $p$ and a positive integer $k$, let$\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions$X \in \mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ to the polynomial equation$P(X) = 0$, where $P(t)$ is a monic polynomial in$(\mathbb{Z}/p^{k}\mathbb{Z})[t]$ whose reduction modulo $p$ is square-freeover the finite field $\mathbb{F}_{p}$ of $p$ elements. Noting that $P(X) = 0$if and only if $\mathrm{cok}(P(X)) \simeq (\mathbb{Z}/p^{k}\mathbb{Z})^{n}$, wegive a conjectural generalization of counting solutions to $P(X) = 0$ as thedistribution of the cokernel $\mathrm{cok}(P(X))$ of $P(X)$ up to isomorphisms,where $X$ is a uniform random matrix in$\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$. This distribution involves anexplicit formula when we fix the residue class of $X$ modulo $p$. We prove thisconjecture for the special case when the image of $P(t)$ in $\mathbb{F}_{p}[t]$modulo $p$ is irreducible. We explain how the distribution we obtain is closelyrelated to the Cohen-Lenstra distribution. Our proof involves algebraic andcombinatorial arguments in linear algebra over $\mathbb{Z}/p^{k}\mathbb{Z}$ andbuilds upon a previous work of Cheong and Kaplan.

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