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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Small solutions to inhomogeneous linear equations over number fields

Authors: Robbin O’Leary and Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 336 (1993), 915-931
MSC: Primary 11D72; Secondary 11H50
MathSciNet review: 1094559
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Abstract: We consider a system of $ M$ independent, inhomogeneous linear equations in $ N > M$ variables having coefficients in an algebraic number field $ k$ . We give a best possible lower bound on the inhomogeneous height of a solution vector in $ {k^N}$ and determine when a solution exists in $ {({\mathcal{O}_S})^N}$, where $ {\mathcal{O}_S}$ is the ring of $ S$-integers in $ k$ . If such a system has a solution vector in $ {({\mathcal{O}_S})^N}$, we show that it has a solution $ \vec \zeta $ in $ {({\mathcal{O}_S})^N}$ such that the inhomogeneous height of $ \vec \zeta $ is relatively small. We give an explicit upper bound for this height in terms of the heights of the matrices defining the linear system. Our method uses geometry of numbers over adele spaces and local to global arguments.

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Article copyright: © Copyright 1993 American Mathematical Society