Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)
Tuesday 5
Patricia Hersch
› 16:00 - 16:30 (30min)
› Djavad Mowafaghian Cinema
The Smith normal form distribution of a random integer matrix
Yinghui Wang  1  , Richard P. Stanley  2  
1 : Department of Mathematics, Columbia University, New York, New York
2 : Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts

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/pswith a prime and 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 μpand determine the density μ for several interesting types of sets. 

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