Exact solution of general integer systems of linear equations

Abstract

Methods are known for the exact computation of the solution of integer systems of linear equations AX = B with a nonsingular coefficient matrix A by congruence techniques. These methods are now generalized for systems with an arbitrary integer coefficient matrix A . To make congruence techniques applicable, a common denominator of all elements of the solution X = A + B must be computed. This is achieved by defining the natural denominator CODE of A + and describing it by some formulas. Methods for the exact computation of additional results (consistency, null space, solution of at most R nonzero elements), a recursive test to save computing time, and a comparison with some results from the literature are presented.