Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A sharp bound for solutions of linear Diophantine equations

Authors: I. Borosh, M. Flahive, D. Rubin and B. Treybig
Journal: Proc. Amer. Math. Soc. 105 (1989), 844-846
MSC: Primary 15A36; Secondary 11D04, 90C10
MathSciNet review: 955458
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Abstract: Let $ Ax = b$ be an $ m \times n$ system of linear equations with rank $ m$ and integer coefficients. Denote by $ Y$ the maximum of the absolute values of the $ m \times m$ minors of the augmented matrix $ \left( {A,b} \right)$. It is proved that if the system has an integral solution, then it has an integral solution $ x = \left( {{x_i}} \right)$ with $ \max \left\vert {{x_i}} \right\vert \leq Y$. The bound is sharp.

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Keywords: Linear equation, integral solutions, minors, rank, bound
Article copyright: © Copyright 1989 American Mathematical Society