An Efficient Version of the Bombieri-Vaaler Lemma

TL;DR

This paper presents a polynomial-time computable form of the Bombieri-Vaaler bound on solutions to linear Diophantine systems, along with a simpler proof, improving computational efficiency and understanding.

Contribution

It introduces a new polynomial-time computable form of the Bombieri-Vaaler bound and provides a simpler proof for it.

Findings

Bound can be computed in polynomial time

Elementary proof simplifies understanding

Enhances practical computation of solutions

Abstract

In their celebrated paper "On Siegel's Lemma", Bombieri and Vaaler found an upper bound on the height of integer solutions of systems of linear Diophantine equations. Calculating the bound directly, however, requires exponential time. In this paper, we present the bound in a different form that can be computed in polynomial time. We also give an elementary (and arguably simpler) proof for the bound.