Counting nonsingular matrices with primitive row vectors

TL;DR

This paper derives an asymptotic formula for counting nonsingular integer matrices with primitive rows, fixed determinant, and bounded norm, and explores the density of such matrices in the space of matrices with a given determinant.

Contribution

It provides the first asymptotic estimates for the number and density of these matrices, linking primitive row vectors to matrix determinants and norms.

Findings

Asymptotic expression for the count of matrices with primitive rows and fixed determinant.

Density asymptotics of matrices with primitive rows in the space of fixed determinant matrices.

Results applicable for large matrix norms and determinants.

Abstract

We give an asymptotic expression for the number of nonsingular integer n-by-n-matrices with primitive row vectors, determinant k, and Euclidean matrix norm less than T, for large T. We also investigate the density of matrices with primitive rows in the space of matrices with determinant k, and determine its asymptotics for large k.