Bounds on Positive Integral Solutions of Linear Diophantine Equations II

Abstract

Let A be an m × n matrix of rank r and B an m × 1 matrix, both with integer entries. Let M 2 be the maximum of the absolute values of the r × r minors of the augmented matrix ( A | B ). Suppose that the system A x = B has a non-trivial solution in non-negative integers. We prove (1) If r = n - 1 then the system A x = B has a non-negative non-trivial solution with entries bounded by M 2 . (2) If A has a r x n submatrix such that none of its r x r minors is 0 and x ≥ 0 is a solution of Ax=B in integers such that is minimal, then .