A class of ABS algorithms for Diophantine linear systems

Abstract

Systems of integer linear (Diophantine) equations arise from various applications. In this paper we present an approach, based upon the ABS methods, to solve a general system of linear Diophantine equations. This approach determines if the system has a solution, generalizing the classical fundamental theorem of the single linear Diophantine equation. If so, a solution is found along with an integer Abaffian (rank deficient) matrix such that the integer combinations of its rows span the integer null space of the cofficient matrix, implying that every integer solution is obtained by the sum of a single solution and an integer combination of the rows of the Abaffian. We show by a counterexample that, in general, it is not true that any set of linearly independent rows of the Abaffian forms an integer basis for the null space, contrary to a statement by Egervary. Finally we show how to compute the Hermite normal form for an integer matrix in the ABS framework.