On systems of equations in free abelian groups

Abstract

In this paper we study the asymptotic probability that a random system of equations in free abelian group $\mathbb{Z}^m$ of rank $m$ is solvable. Denote $SAT(\mathbb{Z}^m, k, n)$ and $SAT_{\mathbb{Q}^m}(\mathbb{Z}^m, k, n)$ the sets of all systems of $n$ equations in $k$ variables in the group $\mathbb{Z}^m$ solvable in $\mathbb{Z}^m$ and $\mathbb{Q}^m$ respectively. We show that asymptotic density of the set $SAT_{\mathbb{Q}^m}(\mathbb{Z}^m, k, n)$ is equal to $1$ for $n \leq k$, and is equal to $0$ for $n > k$. For $n < k$ we give nontrivial estimates for upper and lower asymptotic densities of the set $SAT(\mathbb{Z}^m, k, n)$. When $n > k$ the set $SAT(\mathbb{Z}^m, k, n)$ is negligible. Also for $n \leq k$ we provide some connection between asymptotic density of the set $SAT(\mathbb{Z}^m, k, n)$ and sums over full rank matrices involving their greatest divisors.