A bound on solutions of linear integer equalities and inequalities

Consider a system of linear equalities and inequalities with integer coefficients. We describe the set of rational solutions by a finite generating set of solution vectors. The entries of these vectors can be bounded by the absolute value of a certain subdeterminant. The smallest integer solution of the system has coefficients not larger than this subde- terminant times the number of indeterminates. Up to the latter factor, the bound is sharp. Let A, B, C, D be m x zz-, m x \-,p x n-,p x 1-matrices respectively with integer entries. The rank of A is r, and s is the rank of the (m + p) X n- matrix (c). Let M be an upper bound on the absolute values of those (s — 1) X (s — 1)- or s X i-subdeterminants of the (m + p) X (n + l)-matrix (c d)> which are formed with at least r rows from (A, B). Theorem. If Ax = B and Cx > D have a common integer solution, then they have one with coefficients bounded by (n + \)M. Let Mx, M2, and M3 be upper bounds on the absolute values of the r X r-subdeterminants, the subdeterminants, and the entries of (A, B) respectively. Taking the zz X zz-identity matrix for C and D = 0, we have the following