A bound on solutions of linear integer equalities and inequalities
A bound on solutions of linear integer equalities and inequalities
复制标题
线性整数不等式和不等式解的界限
DOI：
10.1090/s0002-9939-1978-0500555-0
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发表时间：
1978
影响因子：
0.8
通讯作者：
M. Sieveking
中科院分区：
文献类型：
--
作者：
J. Gathen;M. Sieveking
考虑一个线性的相等性和整数系数的不等式。该系统的系数不超过此细分终端的不确定性数量。 1个级别的整数条目。 s - 1）x（s - 1） - 或（m + p）x（n + l）-matrix（c d）>的s x i-subdeterments>，它们至少从（a，b）的至少r行形成。 r x r-子确定剂，亚测定剂和（a，b）的条目分别为zz x zz-sidentity矩阵，我们有以下内容
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