Solving Systems of Linear Diophantine Equations: An Algebraic Approach

We describe through an algebraic and geometrical study, a new method for solving systems of linear diophantine equations. This approach yields an algorithm which is intrinsically parallel. In addition to the algorithm, we give a geometrical interpretation of the satissability of an homogeneous system, as well as upper bounds on height and length of all minimal solutions of such a system. We also show how our results apply to inhomogeneous systems yielding necessary conditions for satissability and upper bounds on the minimal solutions. Solving linear diophantine equations is a problem which appears in many elds, from linear programming to resolution of equations in semigroups or constrained logic programming. In the framework of semigroups, one needs an eecient test for the satissability of equations as well as a way to generate a basis for the set of all solutions. The ee-ciency of the satissability test becomes crucial in constrained equational logic because constraints are accumulated until their conjunction becomes unsatissable. The simplex method is quite convenient for checking the satissability of a set of homogeneous equations , but does not provide an algorithm for nding all solutions. Such algorithms were proposed by G. Huet 9], A. Fortenbacher 6], A. Herold & T. Guckenbiehl 8], M. Clausen & A. Fortenbacher 4] for one equation. In 1987, J.-L. Lambert 10] gave an upper bound on the components of all minimal solutions of a system, and a ner one on the length of minimal solutions of one equation, improving then greatly Huet's algorithm. In 1989, J.-F. Romeuf 14] described an algorithm for solving two equations, using a nite automaton recognizing the solutions. Deenite improvements were then brought in 1989 by E. Contejean and H. Devie 5, 2] who found an extension of Fortenbacher's algorithm to systems of arbitrary size, and L. Pottier who described in 11] a similar algorithm and gave a new upper bound on the length of all minimal solutions of a system. Very recently, L. Pottier gave in 12] another algorithm using Grr obner bases and improved the upper bounds on the minimal solutions.