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The Smith Normal Form Distribution of A Random Integer Matrix

Author(s)
Wang, Yinghui; Stanley, Richard P
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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.
Date issued
2017-01
URI
http://hdl.handle.net/1721.1/116294
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
SIAM Journal on Discrete Mathematics
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Wang, Yinghui and Richard P. Stanley. “The Smith Normal Form Distribution of A Random Integer Matrix.” SIAM Journal on Discrete Mathematics 31, 3 (January 2017): 2247–2268 © 2017 Society for Industrial and Applied Mathematics
Version: Final published version
ISSN
0895-4801
1095-7146

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