Surjectivity of near square random matrices

TL;DR

This paper proves that nearly square random matrices with independent entries are almost certainly surjective over the integer lattice, extending to sparse and dependent matrices, thus answering a longstanding open question.

Contribution

It establishes the surjectivity of nearly square iid random integral matrices over the integer lattice, including sparse and dependent cases, advancing understanding in random matrix theory.

Findings

High probability of surjectivity for nearly square iid matrices

Extension of results to sparse matrices

Extension of results to matrices with dependent entries

Abstract

We show that a nearly square iid random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz. Our result extends to sparse matrices as well as to matrices of dependent entries.