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 $<p^t$. 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 $t\ge 3$. 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.

title page, plus 11 pages, no illustrations, submitted to a conference

Cited by in corpus (1)

• Randomized Polynomial-Time Root Counting in Prime Power Rings