Reduced normal form of local integral bases

TL;DR

This paper presents a canonical reduced normal form for bases of integral closures of discrete valuation rings, along with an algorithm for computing it, enhancing applications like object identification and basis construction.

Contribution

It introduces a new reduced normal form for local integral bases and provides an algorithm to compute it, improving upon Hermite normal form limitations.

Findings

The normal form facilitates isomorphism identification.

The algorithm efficiently computes reduced bases.

Reduced bases are crucial for key applications.

Abstract

We introduce a canonical form for reduced bases of integral closures of discrete valuation rings, and we describe an algorithm for computing a basis in reduced normal form. This normal form has the same applications as the Hermite normal form: identification of isomorphic objects, construction of global bases by patching local ones, etc. but in addition the bases are reduced, which is a crutial property for several important applications. Except for very particular cases, a basis in Hermite normal form cannot be reduced.