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Proceedings of the American Mathematical Society
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0955458-1
PII: S 0002-9939(1989)0955458-1
Keywords: Linear equation, integral solutions, minors, rank, bound
Article copyright: © Copyright 1989 American Mathematical Society