Sparse solutions of linear Diophantine equations

Abstract

We present structural results on solutions to the Diophantine system $A\mathbf{y}=\mathbf{b}$, $\mathbf{y}\in {\mathbb{Z}}_{\ge 0}^t$ 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.

References in corpus (1)

• On factorization invariants and Hilbert functions

Cited by in corpus (6)

• Sparsity of integer solutions in the average case
• Augmented Hilbert series of numerical semigroups
• Improving proximity bounds using sparsity
• Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones
• Minkowski's successive minima in convex and discrete geometry
• Polynomial upper bounds on the number of differing columns of $\Delta$-modular integer programs