F_p is locally like C

Abstract

Vu, Wood and Wood showed that any finite set S in a characteristic zero integral domain can be mapped to F_p, for infinitely many primes p, while preserving finitely many algebraic incidences of S. In this note we show that the converse essentially holds, namely any small subset of F_p can be mapped to some finite algebraic extension of Q, while preserving bounded algebraic relations. This answers a question of Vu, Wood and Wood. We give several applications, in particular we show that for small subsets of F_p, the Szemer\'edi-Trotter theorem holds with optimal exponent 4/3, and we improve the previously best-known sum-product estimate in F_p. We also give an application to an old question of R\'enyi. The proof of the main result is an application of elimination theory and is similar in spirit with the proof of the quantitative Hilbert Nullstellensatz.