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arXiv · 1506.00160

The Smith Normal Form Distribution of a Random Integer Matrix

Yinghui Wang·Richard P. Stanley
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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.

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Yinghui Wang, Richard P. Stanley. 2015-05-30. The Smith Normal Form Distribution of a Random Integer Matrix. https://doi.org/10.1137/16m1098140

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