Natural Density Distribution of Hermite Normal Forms of Integer Matrices

TL;DR

This paper explicitly computes the asymptotic distribution of Hermite Normal Forms of large integer matrices with bounded entries, revealing non-uniformity and enabling analysis of related algorithms.

Contribution

It provides an explicit asymptotic formula for the distribution of HNFs of integer matrices with bounded entries, a novel contribution to understanding their probabilistic structure.

Findings

Distribution of HNFs is far from uniform for large matrices.

Explicit asymptotic counts for matrices with prescribed HNF structures.

Applications to analyzing algorithms dependent on HNF properties.

Abstract

The Hermite Normal Form (HNF) is a canonical representation of matrices over any principal ideal domain. Over the integers, the distribution of the HNFs of randomly looking matrices is far from uniform. The aim of this article is to present an explicit computation of this distribution together with some applications. More precisely, for integer matrices whose entries are upper bounded in absolute value by a large bound, we compute the asymptotic number of such matrices whose HNF has a prescribed diagonal structure. We apply these results to the analysis of some procedures and algorithms whose dynamics depend on the HNF of randomly looking integer matrices.