A SHARP BOUND ON POSITIVE SOLUTIONS OF LINEAR DIOPHANTINE EQUATIONS

Let Ax = b be an m x n system of linear equations with rank m < n and integer coefficients. Denote by Y the maximum of the absolute values of the m x m minors of the augmented matrix (A, b). It is proved that if the system has an integral solution x = (x(i)) with each x(i) greater-than-or-equal-to 0, and either Ax = 0 has no such solution which is nontrivial or there is an m x (m + 1) submatrix A' of A with rank m such that A' y = 0 has a solution with positive integer components, then Ax = b has an integral solution with each 0 less-than-or-equal-to x(i) less-than-or-equal-to Y. The bound is sharp.