Counting Roots of Polynomials Over Prime Power Rings

Abstract: Suppose $p$ is a prime, $t$ is a positive integer, and $f!\in!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $&lt;!p<sup>t$. We show that for any fixed $t$, we can compute the number of roots in $\mathbb{Z}/(p<sup>t)$ of $f$ in deterministic time $(d+\log p)<sup>{O(1)}$. This fixed parameter tractability appears to be new for $t!\geq!3$. 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.