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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Diophantine approximation by algebraic hypersurfaces and varieties


Author: Wolfgang M. Schmidt
Journal: Trans. Amer. Math. Soc. 359 (2007), 2221-2241
MSC (2000): Primary 11J13, 11J81, 11J82, 11J85
Published electronically: December 5, 2006
MathSciNet review: 2276618
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Abstract | References | Similar Articles | Additional Information

Abstract: Questions on rational approximations to a real number can be generalized in two directions. On the one hand, we may ask about ``approximation'' to a point in $ \mathbb{R}^{n}$ by hyperplanes defined over the rationals. That is, we seek hyperplanes with small distance from the given point. On the other hand, following Wirsing, we may ask about approximation to a real number by real algebraic numbers of degree at most $ d$.

The present paper deals with a common generalization of both directions, namely with approximation to a point in $ \mathbb{R}^{n}$ by algebraic hypersurfaces, or more generally algebraic varieties defined over the rationals.


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Additional Information

Wolfgang M. Schmidt
Affiliation: Department of Mathematics, University of Colorado, Boulder, Colorado 80309-0395

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04014-1
PII: S 0002-9947(06)04014-1
Keywords: Simultaneous approximation, hypersurfaces, algebraic independence
Received by editor(s): October 12, 2004
Received by editor(s) in revised form: March 10, 2005
Published electronically: December 5, 2006
Additional Notes: The author was partially supported by NSF DMS 0074531
Article copyright: © Copyright 2006 American Mathematical Society