Fp 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 Szemerédi–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 Rényi. The proof of the main result is an application of elimination theory and is similar in spirit to the proof of the quantitative Hilbert Nullstellensatz.