A polynomial time algorithm for computing the HNF of a module over the integers of a number field

TL;DR

This paper introduces a polynomial time algorithm for computing the Hermite Normal Form of modules over the ring of integers in a number field, addressing previous conjectures about its efficiency.

Contribution

The paper provides a new method to control coefficient growth and rigorously proves the polynomial complexity of the algorithm for modules over number field integers.

Findings

Algorithm runs in polynomial time with respect to input size

New method prevents coefficient explosion

Complexity is assessed based on field invariants

Abstract

We present a variation of the modular algorithm for computing the Hermite Normal Form of an $\OK$-module presented by Cohen, where $\OK$ is the ring of integers of a number field K. The modular strategy was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we provide a new method to prevent the coefficient explosion and we rigorously assess its complexity with respect to the size of the input and the invariants of the field K.