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28 March 2024
 
  » arxiv » 1303.2363

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F_p is locally like C
Codrut Grosu ;
Date 10 Mar 2013
AbstractVu, 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. 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.
Source arXiv, 1303.2363
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