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 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.
Highlights
Let M be a nonzero n × m matrix over a commutative ring R, and r be the rank of M
If there exist invertible n × n and m × m matrices P and Q such that the product P M Q is a diagonal matrix with diagonal entries d1, d2, . . . , dr, 0, 0, . . . , 0 satisfying that di | di+1 for all 1 ≤ i ≤ r − 1, P M Q is the Smith normal form (SNF) of M
The diagonal entries are uniquely determined by gi−1di = gi (1 ≤ i ≤ r), where g0 = 1 and gi is the greatest common divisor of all i × i minors of M
Summary
Let M be a nonzero n × m matrix over a commutative ring R (with identity), and r be the rank of M. Note that the special case that s = 0 or 1 follows from Ekedahl (1991, Theorem 2.3) directly This result applies to the probability that g(x) = 1 , in other words, that the polynomial values are relatively prime. All these results hold for the multi-gcd distribution of polynomial values, namely, when g(x) is a vector whose components are the gcds of the values of given collections of polynomials at x We apply this theory to the SNF distribution of a random integer matrix to show that the density μ (of SNF of a random n × m integer matrix) equals a product of some densities μps of SNF over Z/psZ for sets of form (3.2) (Theorem 3.5). We shall assume throughout that p represents a prime, pj is the j-th smallest prime, and p means a product over all primes p
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