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F_p is locally like C | Codrut Grosu
; | Date: |
10 Mar 2013 | 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. 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 | Services: | Forum | Review | PDF | Favorites |
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