Computing Puiseux series : a fast divide and conquer algorithm

TL;DR

This paper introduces a fast divide and conquer algorithm for computing Puiseux series of algebraic curves, improving efficiency in factorization, irreducibility testing, and genus computation over various fields.

Contribution

It presents a novel divide and conquer approach that replaces univariate factorization with dynamic evaluation, enabling faster computation of Puiseux series, irreducible factors, and genus of algebraic curves.

Findings

Computes Puiseux series singular parts in less than (Delta) operations.

Factors of F up to X^N can be computed in (D(elta + N)) operations.

Genus of the curve can be computed in (D^3) operations, with probabilistic methods over Q.

Abstract

Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the singular parts of all Puiseux series of $F$ above $X = 0$ in less than $\tilde{\mathcal{O}}(D\delta)$ operations in $\mathbb{K}$, where $\delta$ is the valuation of the resultant of $F$ and its partial derivative with respect to $Y$. To this aim, we use a divide and conquer strategy and replace univariate factorization by dynamic evaluation. As a first main corollary, we compute the irreducible factors of $F$ in $\mathbb{K}[[X]][Y ]$ up to an arbitrary precision $X^N$ with $\tilde{\mathcal{O}}(D(\delta + N ))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by $F$ with $\tilde{\mathcal{O}}(D^3)$ arithmetic operations and, if $\mathbb{K} = \mathbb{Q}$, with $\tilde{\mathcal{O}}((h+1)D^3)$ bit operations using a probabilistic algorithm, where $h$ is the logarithmic heigth of $F$.