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 $SA{T}_{{\mathbb{Q}}^m}({\mathbb{Z}}^m,k,n)$ 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 $SA{T}_{{\mathbb{Q}}^m}({\mathbb{Z}}^m,k,n)$ is equal to $1$ for $n\le k$, 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 $n\le k$ 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.

15 pages

References in corpus (2)

• On the asymptotics of visible elements and homogeneous equations in surface groups
• Asymptotic density of rational sets in free abelian groups