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

MathSciNet review:
3223350

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Abstract | References | Similar Articles | Additional Information

Abstract: The celebrated Artin-Whaples approximation theorem (which is a generalization of the Chinese remainder theorem) asserts that, given a field , distinct places of , and points , it is possible to find an simultaneously near w.r.t. with any prescribed accuracy. If we replace with other algebraic varieties , the analogous conclusion does not generally hold, e.g., because may contain too few points over . However, it has been proved by a number of authors that, at least in the case of global fields, it holds if we allow to be algebraic over . These results do not directly contain either the case of or the case of general fields, and above all they do not control the degree of .

In this paper we offer different arguments leading to a general approximation theorem properly generalizing that of Artin-Whaples. This works for every , as above, and not only asserts the existence of a suitable , but bounds explicitly the degree in terms only of geometric invariants of . 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

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