On rectangular unimodular matrices over the algebraic integers

TL;DR

This paper calculates the density of rectangular unimodular matrices over algebraic integers in a number field, extending understanding of their distribution in algebraic number theory.

Contribution

It provides the first explicit computation of the density of such matrices over algebraic integers, generalizing previous results over simpler rings.

Findings

Explicit density formulas derived for matrices over algebraic integers

Extension of unimodular matrix distribution results to algebraic number fields

New insights into the structure of unimodular matrices in algebraic settings

Abstract

Let $n$ and $m$ be positive integers such that $n<m$. In this paper we compute the density of rectangular unimodular $n$ by $m$ matrices over the ring of algebraic integers of a number field.