Small solutions of linear Diophantine equations

Let Ax = B be a system of m x n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m x m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = (xi) in nonnegative integers with xi <~ X for n m variables and xi ~< (n m + 1)Y for m variables. This improves previous results of the authors and others.