Nonnegative elements of subgroups of \(\mathbb{Z}^ n\) (Q1377517)

Let \(Ax=0\) be a homogeneous system of linear equations with integer coefficients. There exists a set of solutions \(\{s_1,\dots,s_t\}\) such that every nonnegative integer solution can be expressed as a linear combination of \(\{s_i\}\) with nonnegative integer coefficients. It is shown in this paper that the elements \(s_1,\dots,s_t\) can be found in the set of integer vectors \((x_1,\dots,x_n)\) such that \(x_1+\dots+x_n\leq (r-1)(n-r)D\). This method can also be used to solve the problem of finding nonnegative integer solutions of homogeneous systems of linear equations with integer coefficients, where some or all of the equations are in the form of congruences.