# Fast Computation of the Roots of Polynomials Over the Ring of Power Series

## TL;DR

This paper presents an efficient algorithm for finding all roots of polynomials over power series rings, handling multiple roots and improving computational complexity compared to previous methods.

## Contribution

It introduces a divide & conquer algorithm with optimal cost bounds, extending root-finding to multiple roots over power series rings.

## Key findings

- Algorithm computes roots with cost proportional to input size
- Handles multiple roots where Newton iteration fails
- Improves complexity over previous quadratic algorithms

## Abstract

We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field $\mathbb{K}$. More precisely, given a precision $d$, and a polynomial $Q$ whose coefficients are power series in $x$, the algorithm computes a representation of all power series $f(x)$ such that $Q(f(x)) = 0 \bmod x^d$. The algorithm works unconditionally, in particular also with multiple roots, where Newton iteration fails. Our main motivation comes from coding theory where instances of this problem arise and multiple roots must be handled. The cost bound for our algorithm matches the worst-case input and output size $d \deg(Q)$, up to logarithmic factors. This improves upon previous algorithms which were quadratic in at least one of $d$ and $\deg(Q)$. Our algorithm is a refinement of a divide \& conquer algorithm by Alekhnovich (2005), where the cost of recursive steps is better controlled via the computation of a factor of $Q$ which has a smaller degree while preserving the roots.