The corank of a rectangular random integer matrix

TL;DR

This paper investigates the probability that large rectangular random integer matrices are surjective on integer lattices, establishing high-probability results for certain dimensions and discussing conjectures and counterexamples for narrower matrices.

Contribution

It proves that under certain conditions, large random integer matrices are surjective with high probability and discusses conjectures and necessary conditions for narrower matrices.

Findings

Large $n \times (2+\epsilon)n$ matrices are surjective with probability $1 - O(e^{-cn})$.

Counterexamples show the necessity of the conditions for surjectivity.

Conjecture that narrower matrices $n \times (1+\epsilon)n$ are also surjective under similar conditions.

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.