On the Smallest Support Size of Integer Solutions to Linear Equations

TL;DR

This paper establishes an asymptotically tight upper bound on the minimal support size of integer solutions to linear equations, simplifying previous proofs and applicable in worst-case scenarios.

Contribution

It provides a simpler proof for the bound on support size of integer solutions, matching the best known bounds and focusing on worst-case analysis.

Findings

Bound on support size is asymptotically tight

Proof relies solely on linear algebra

Matches previous bounds from 2022

Abstract

In this note, we study the size of the support of integer solutions to linear equations $Ax=b, ~x\in\Z^n$ where $A\in\Z^{m\times n}, b\in\Z^n$. We give an upper bound on the smallest support size as a function of $A$, taken as a worst case over all $b$ such that the above system has a solution. This bound is asymptotically tight, and in fact matches the bound given in Aliev, Averkov, De Loera and Oertel Mathematical Programming 2022, while the proof presented here is simpler, relying only on linear algebra.