The Smith normal form distribution of a random integer matrix

Yinghui Wang (1) , Richard P. Stanley (2)

1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, United States
2 MIT - Massachusetts Institute of Technology

We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/psZ 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 determine the density μ for several interesting types of sets.

Mots clés: Combinatorics

Domaines: Combinatoire [math.CO]

Yinghui Wang, Richard P. Stanley. The Smith normal form distribution of a random integer matrix. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6352⟩. ⟨hal-02166330⟩