Random systems of equations in free abelian groups

Abstract

We study the solvability of random systems of equations on the free abelian group ℤ^m of rank m. Denote by SAT(ℤ^m, k, n) and $SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)$ the sets of all systems of n equations of k unknowns in ℤ^m satisfiable in ℤ^m and ℚ^m respectively. We prove that the asymptotic density $\rho \left( {SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)} \right)$ of the set $SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)$ equals 1 for n ≤ k and 0 for n > k. As regards, SAT(ℤ^m, k, n) for n < k, some new estimates are obtained for the lower and upper asymptotic densities and it is proved that they lie between (Π ^k _j=k−n+1 ζ(j))⁻¹ and $\left( {\tfrac{{\zeta (k + m)}} {{\zeta (k)}}} \right)^n$, where ξ(s) is the Riemann zeta function. For n ≤ k, a connection is established between the asymptotic density of SAT(ℤ^m, k, n) and the sums of inverse greater divisors over matrices of full rank. Starting from this result, we make a conjecture about the asymptotic density of SAT(ℤ^m, n, n). We prove that ρ(SAT(ℤ^m, k, n)) = 0 for n > k.