The Smith Normal Form Distribution of a Random Integer Matrix

TL;DR

This paper proves the existence of a density distribution for the Smith normal form of a random integer matrix, connecting it to prime power local densities and analyzing its properties.

Contribution

It introduces a novel connection between SNF distribution and multi-gcd distributions of polynomial values, providing explicit formulas and analyzing density properties.

Findings

Density of SNF exists and equals a product of local densities.

Derived explicit formulas for local densities $mma_{p^s}$.

Analyzed the maximum, minimum, and monotonicity of the densities.

Abstract

We show that the density $\mu$ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities $\mu_{p^s}$ of SNF over $\mathbb{Z}/p^s\mathbb{Z}$ with $p$ a prime and $s$ some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for $\mu_{p^s}$ and compute the density $\mu$ for several interesting types of sets. Finally, we determine the maximum and minimum of $\mu_{p^s}$ and establish its monotonicity properties and limiting behaviors.