A Discrete Farkas Lemma

Given A ∈ Zm×n and b ∈ Zm, we consider the issue of existence of a nonnegative integral solution x ∈ Nn to the system of linear equations Ax = b. We provide a discrete and explicit analogue of the celebrated Farkas lemma for linear systems in Rn and prove that checking existence of integral solutions reduces to solving an explicit linear programming problem of fixed dimension, known in advance.