Sparse solutions of linear Diophantine equations

TL;DR

This paper investigates the structure of sparse solutions to linear Diophantine equations, using algebraic and number theoretic tools, with implications for discrete optimization.

Contribution

It provides new structural results on minimal-support solutions to linear Diophantine systems using algebraic and number theoretic methods.

Findings

Characterization of solutions with minimal non-zero entries

Application of Siegel's Lemma and generating functions

Implications for discrete optimization problems

Abstract

We present structural results on solutions to the Diophantine system $A{\boldsymbol y} = {\boldsymbol b}$, ${\boldsymbol y} \in \mathbb Z^t_{\ge 0}$ with the smallest number of non-zero entries. Our tools are algebraic and number theoretic in nature and include Siegel's Lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.