Integral matrices of fixed rank

Abstract

Asymptotic formulæare derived for the number of n × m n \times m matrices of fixed rank k k with rational integral coefficients that are contained in a Euclidean ball of radius T T in R n × m {{\mathbf {R}}^{n \times m}} . It is assumed that n ⩾ m > k ⩾ 1 n \geqslant m > k \geqslant 1 are fixed, and the asymptotics are valid as T T tends to infinity. The methods used are elementary.