# Sum–product estimates for rational functions

@article{Bukh2012SumproductEF,
title={Sum–product estimates for rational functions},
author={Boris Bukh and Jacob Tsimerman},
journal={Proceedings of the London Mathematical Society},
year={2012},
volume={104}
}
• Published 12 February 2010
• Mathematics
• Proceedings of the London Mathematical Society
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 𝔽p. 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…
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