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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Artin-Whaples approximations of bounded degree in algebraic varieties


Authors: Vincenzo Mantova and Umberto Zannier
Journal: Proc. Amer. Math. Soc. 142 (2014), 2953-2964
MSC (2010): Primary 11G99
Published electronically: May 1, 2014
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Abstract: The celebrated Artin-Whaples approximation theorem (which is a generalization of the Chinese remainder theorem) asserts that, given a field $ K$, distinct places $ v_{1},\dots ,v_{n}$ of $ K$, and points $ x_{1},\dots ,x_{n}\in \mathbb{P}_{1}(K)$, it is possible to find an $ x\in \mathbb{P}_{1}(K)$ simultaneously near $ x_{i}$ w.r.t. $ v_{i}$ with any prescribed accuracy. If we replace $ \mathbb{P}_{1}$ with other algebraic varieties $ V$, the analogous conclusion does not generally hold, e.g., because $ V$ may contain too few points over $ K$. However, it has been proved by a number of authors that, at least in the case of global fields, it holds if we allow $ x$ to be algebraic over $ K$. These results do not directly contain either the case of $ \mathbb{P}_{1}$ or the case of general fields, and above all they do not control the degree of $ x$.

In this paper we offer different arguments leading to a general approximation theorem properly generalizing that of Artin-Whaples. This works for every $ V$, $ K$ as above, and not only asserts the existence of a suitable $ x\in V(\overline {K})$, but bounds explicitly the degree $ [K(x):K]$ in terms only of geometric invariants of $ V$. It shall also be seen that such a bound is in a sense close to being best-possible.


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Additional Information

Vincenzo Mantova
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Address at time of publication: Scuola di Scienze e Tecnologie, Sezione di Matematica, Università degli Studi di Camerino, Via Madonna delle Carceri 9, 62032 Camerino (MC), Italy
Email: vincenzo.mantova@unicam.it

Umberto Zannier
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Email: u.zannier@sns.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-12021-1
PII: S 0002-9939(2014)12021-1
Received by editor(s): March 8, 2012
Received by editor(s) in revised form: September 7, 2012
Published electronically: May 1, 2014
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society