Sum-product estimates for rational functions
Combinatorics
2014-02-26 v2 Algebraic Geometry
Number Theory
Abstract
We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is given for the rational functions f and g for which |f(A)+f(A)|+|g(A,A)| can be as small as |A|, for large |A|. Third, we show that under mild conditions on f, |f(A,A)| is substantially larger than |A|, provided |A| is large. We also present a conjecture on what the general sum-product result should be.
Cite
@article{arxiv.1002.2554,
title = {Sum-product estimates for rational functions},
author = {Boris Bukh and Jacob Tsimerman},
journal= {arXiv preprint arXiv:1002.2554},
year = {2014}
}
Comments
32 pages, small additions, several typos fixed