The Smith Normal Form Distribution of A Random Integer Matrix

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 $\Bbb{Z}/p^s\Bbb{Z}$ 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. As an application, we determine the probability that the cokernel of a random integer square matrix has at most $\ell$ generators for a positive integer $\ell$, and establish its asymptotics as $\ell \to \infty$, which extends a result of Ekedahl (1991) on the case $\ell=1$.