ARTIN-WHAPLES APPROXIMATIONS OF BOUNDED DEGREE IN ALGEBRAIC VARIETIES

作者：Mantova Vincenzo^*; Zannier Umberto

来源：PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2014, 142(9): 2953-2964.

DOI：10.1090/S0002-9939-2014-12021-1

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), ... ,v(n) of K, and points x(1), ... ,x(n) is an element of P-1 (K), it is possible to find an x is an element of P-1 (K) simultaneously near x(i) w.r.t. v(i) with any prescribed accuracy. If we replace 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 P-1 or the case of general fields, and above all they do not control the degree of x. %26lt;br%26gt;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 is an element of V ((K) over bar), 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.

• 出版日期2014-9