Counting Roots of Polynomials Over Prime Power Rings

TL;DR

This paper presents a deterministic algorithm for counting roots of univariate polynomials over prime power rings, achieving fixed parameter tractability for any fixed prime power exponent, with applications to computing Igusa zeta functions.

Contribution

It introduces a new fixed parameter tractable method for root counting over prime power rings for any fixed exponent, which was previously unknown for exponents greater than or equal to 3.

Findings

Efficient root counting algorithm with polynomial time complexity for fixed t

Applicable to computing Igusa zeta functions for fixed degree

Advances the understanding of polynomial root structure 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 $<\!p^t$. We show that for any fixed $t$, we can compute the number of roots in $\mathbb{Z}/(p^t)$ of $f$ in deterministic time $(d+\log p)^{O(1)}$. 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.