Triangular bases of integral closures

TL;DR

This paper introduces the MaxMin algorithm for efficiently computing triangular bases of integral closures of one-dimensional local rings, outperforming existing methods in most practical scenarios.

Contribution

The paper presents a new algorithm, MaxMin, for computing triangular bases of integral closures, with theoretical complexity comparable to current methods but improved practical performance.

Findings

MaxMin algorithm is faster than standard computer algebra routines in most cases.

Theoretical complexity matches state-of-the-art methods.

Practical efficiency is significantly improved, especially for larger extensions.

Abstract

In this work, we consider the problem of computing triangular bases of integral closures of one-dimensional local rings. Let $(K, v)$ be a discrete valued field with valuation ring $\mathcal{O}$ and let $\mathfrak{m}$ be the maximal ideal. We take $f \in \mathcal{O}[x]$, a monic irreducible polynomial of degree $n$ and consider the extension $L = K[x]/(f(x))$ as well as $\mathcal{O}_{L}$ the integral closure of $\mathcal{O}$ in $L$, which we suppose to be finitely generated as an $\mathcal{O}$-module. The algorithm $\operatorname{MaxMin}$, presented in this paper, computes triangular bases of fractional ideals of $\mathcal{O}_{L}$. The theoretical complexity is equivalent to current state of the art methods and in practice is almost always faster. It is also considerably faster than the routines found in standard computer algebra systems, excepting some cases involving very small field extensions.