Counting Roots of Polynomials Over Prime Power Rings

Suppose $p$ is a prime, $t$ is a positive integer, and $f\text{\hspace{0.17em}⁣}\in \text{\hspace{0.17em}⁣}\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\text{\hspace{0.17em}⁣}{p}^t$. We show that for any fixed $t$, we can compute the number of roots in $\mathbb{Z}\mathrm{/}(p^t)$ of $f$ in deterministic time $(d+\log p{)}^{O(1)}$. This fixed parameter tractability appears to be new for $t\text{\hspace{0.17em}⁣}\ge \text{\hspace{0.17em}⁣}3$. A consequence for arithmetic geometry is that we can efficiently compute Igusa zeta functions $Z$, for univariate polynomials, assuming the degree of $Z$ is fixed.