Artin-Whaples approximations of bounded degree in algebraic varieties

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.