The corank of a rectangular random integer matrix

Abstract

We show that under reasonable conditions, a random $n\times (2+\epsilon )n$ integer matrix is surjective on ${\mathbb{Z}}^n$ with probability $1-O(e^{-cn})$. We also conjecture that this should hold for $n\times (1+\epsilon )n$, and provide a counterexample to show that our "reasonableness" conditions are necessary.