The Smith Normal Form Distribution of A Random Integer Matrix

Abstract

We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μ p of SNF over ℤ/p s 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 μps and compute the density μ 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 ℓ generators for a positive integer ℓ, and establish its asymptotics as ℓ → ∞, which extends a result of Ekedahl (1991) on the case ℓ = 1.