Counting Roots of Polynomials over $\mathbb{Z}/p^2\mathbb{Z}$

TL;DR

This paper introduces a new efficient algorithm for counting roots of polynomials over the ring rac{p^2}{Z} in polynomial time, overcoming previous brute-force limitations and providing a concise root-count formula.

Contribution

It presents the first polynomial-time algorithm for counting roots of polynomials over rac{p^2}{Z}, with a new formula for the number of roots.

Findings

Algorithm runs in polynomial time in degree, size, and rac{p}{log p}

Provides a concise formula for root count over rac{p^2}{Z}

Advances root-finding methods over prime power rings

Abstract

Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors. Fix a prime $p \in \mathbb{Z}$ and $f \in ( \mathbb{Z}/p^n \mathbb{Z} ) [x]$ any nonzero polynomial of degree $d$ whose coefficients are not all divisible by $p$. For the case $n=2$, we prove a new efficient algorithm to count the roots of $f$ in $\mathbb{Z}/p^2\mathbb{Z}$ within time polynomial in $(d+\operatorname{size}(f)+\log{p})$, and record a concise formula for the number of roots, formulated by Cheng, Gao, Rojas, and Wan.