Computing integral bases via localization and Hensel lifting

TL;DR

This paper introduces a novel, parallelizable algorithm for computing integral bases in algebraic function fields using localization, Hensel lifting, and Puiseux expansions, outperforming previous methods.

Contribution

The new algorithm reduces reliance on Newton-Puiseux expansions by employing Hensel's lemma, enabling more efficient and parallel computation of integral bases for algebraic curves.

Findings

Outperforms existing algorithms in most cases

Uses Hensel's lemma to avoid full Puiseux expansion computation

Enables parallel processing for faster computation

Abstract

We present a new algorithm for computing integral bases in algebraic function fields of one variable, or equivalently for constructing the normalization of a plane curve. Our basic strategy makes use of the concepts of localization and completion, together with the Chinese remainder theorem, to reduce the problem to the task of finding integral bases for the branches of each singularity of the curve. To solve the latter task, in turn, we work with suitably truncated Puiseux expansions. In contrast to van Hoeij's algorithm, which also relies on Puiseux expansions (but pursues a different strategy), we use Hensel's lemma as a key ingredient. This allows us at some steps of the algorithm to compute factors corresponding to conjugacy classes of Puiseux expansions, without actually computing the individual expansions. In this way, we make substantially less use of the Newton-Puiseux algorithm. In addition, our algorithm is inherently parallel. As a result, it outperforms in most cases any other algorithm known to us by far. Typical applications are the computation of adjoint ideals and, based on this, the computation of Riemann-Roch spaces and the parametrization of rational curves.