Bounds on Positive Integral Solutions of Linear Diophantine Equations

Assuming the existence of a solution, we find bounds for small solutions x of the finite matrix equation Ax = B, where each entry of A, B is an integer, and x is a nontrivial column vector with nonnegative integer entries. 0. Introduction. In [1], [5] and [6] there arise in a topological setting, systems of linear equations with integer coefficients. The problem is to find a bound K depending on the coefficients only, such that if the given system has a nontrivial solution in nonnegative integers, then it has such a solution with all entries bounded by K. In [6] L. B. Treybig gives such a bound K using an inductive definition, and proves that a new solution can be found bounded by K and with the additional property that each entry is bounded by the corresponding entry of the given solution. The purpose of this paper is to find some easily stated bounds which are much smaller than those given by Treybig [6] but which do not satisfy necessarily the additional property. Notation. Throughout this paper A will denote an m X n matrix, B an m x 1 matrix, both with integral entries. We will consider the system of equations (1) Ax = B where x is a column whose entries are x,, . . . , xn. By (A\B) we denote the augmented matrix of the system (1). Let r denote the rank of A, Mx the maximum of the absolute values of all the minors of A of order r, M2 the maximum of the absolute values of all minors of order r of (A\B), and M the maximum of the absolute values of all the minors of {A\B). All our bounds will be stated in terms of m, «, Mx, M2, M, and therefore we may assume from now on without loss of generality that r = m < «. [x] will denote, as usual, the largest integer not exceeding x. Results. §1 is devoted to the case m = 1. We prove that the bound in this case is the maximum of the absolute value of the coefficients. It is easy to see that this bound is sharp. §2 considers the case r = n — 1 and we find the bound M2{1 + l/Mx). The homogeneous case, 5 = 0, is discussed in §3 and the bound M is obtained. Received by the editors June 21, 1974, and, in revised form, March 10, 1975. AMS (MOS) subject classifications (1970). Primary 15A06; Secondary 10B05.