The Smith Normal Form Distribution of a Random Integer Matrix

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 $\mathbb{Z}\mathrm{/}{p}^s\mathbb{Z}$ 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.