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

Given a prime $p$ and a positive integer $k$, let ${\mathrm{M}}_n(\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z})$ be the ring of $n\times n$ matrices over $\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z}$. We consider the number of solutions $X\in {\mathrm{M}}_n(\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z})$ to the polynomial equation $P(X)=0$, where $P(t)$ is a monic polynomial in $(\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z})[t]$ whose reduction modulo $p$ is square-free over the finite field ${\mathbb{F}}_p$ of $p$ elements. Noting that $P(X)=0$ if and only if $\mathrm{c}\mathrm{o}\mathrm{k}(P(X))\simeq (\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z}{)}^n$, we give a conjectural generalization of counting solutions to $P(X)=0$ as the distribution of the cokernel $\mathrm{c}\mathrm{o}\mathrm{k}(P(X))$ of $P(X)$ up to isomorphisms, where $X$ is a uniform random matrix in ${\mathrm{M}}_n(\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z})$. 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 ${\mathbb{F}}_p[t]$ modulo $p$ is irreducible. We explain how the distribution we obtain is closely related to the Cohen-Lenstra distribution. Our proof involves algebraic and combinatorial arguments in linear algebra over $\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z}$ and builds upon a previous work of Cheong and Kaplan.