Sparse Solutions of Linear Diophantine Equations

Abstract

We present structural results on solutions to the Diophantine system $Ay = b$, $ y \in \mathbb{Z}^t_{\ge 0}$ that have the smallest number of nonzero 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.