Randomized Polynomial-Time Root Counting in Prime Power Rings

TL;DR

This paper introduces a randomized polynomial-time algorithm for counting roots of univariate polynomials over prime power rings, significantly improving upon previous exponential-time methods and demonstrating practical potential.

Contribution

It provides the first efficient randomized algorithm for root counting in prime power rings with polynomial complexity, surpassing prior exponential-time approaches.

Findings

Algorithm runs in polynomial time with respect to degree and prime power size.

Experimental data suggests the algorithm's practical applicability.

Complexity bounds are slightly better than initial theoretical estimates.

Abstract

Suppose $k,p\!\in\!\mathbb{N}$ with $p$ prime and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the number of roots of $f$ in $\mathbb{Z}/\!\left(p^k\right)$ within time $d^3(k\log p)^{2+o(1)}$. (We in fact prove a more intricate complexity bound that is slightly better.) The best previous general algorithm had (deterministic) complexity exponential in $k$. We also present some experimental data evincing the potential practicality of our algorithm.