Yinghui Wang ; Richard P. Stanley

The Smith normal form distribution of a random integer matrix
dmtcs:6352 
Discrete Mathematics & Theoretical Computer Science,
April 22, 2020,
DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)

https://doi.org/10.46298/dmtcs.6352
The Smith normal form distribution of a random integer matrixArticle
Authors: Yinghui Wang ^{1}; Richard P. Stanley ^{2}
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Yinghui Wang;Richard P. Stanley
1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, United States
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 multigcd 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.