Natural Density of Rectangular Unimodular Integer Matrices

TL;DR

This paper calculates the natural density of rectangular integer matrices that can be extended to invertible matrices, linking number theory and algebraic geometry concepts.

Contribution

It provides a new explicit formula for the density of such matrices and explores connections with classical theorems like Cesaro's and Quillen-Suslin's.

Findings

Derived the density formula for extendable matrices.

Connected matrix density with coprimality and algebraic theorems.

Identified the density of matrices with Hermite normal form [O Id].

Abstract

In this paper, we compute the natural density of the set of k x n integer matrices that can be extended to an invertible n x n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form [O Id]. Connections with Cesaro's Theorem on the density of coprime integers and Quillen-Suslin's Theorem are also presented.