  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.
