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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Small solutions to inhomogeneous linear equations over number fields
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by Robbin O’Leary and Jeffrey D. Vaaler PDF
Trans. Amer. Math. Soc. 336 (1993), 915-931 Request permission

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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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 336 (1993), 915-931
  • MSC: Primary 11D72; Secondary 11H50
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1094559-2
  • MathSciNet review: 1094559