Sum-product estimates for rational functions

We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, vertical bar f(A) + f(A)vertical bar + vertical bar AA vertical bar is substantially larger than vertical bar A vertical bar for an arbitrary polynomial f over F-p. Second, a characterization is given for the rational functions f and g for which vertical bar f(A) + f(A)vertical bar + vertical bar g(A, A)vertical bar can be as small as vertical bar A vertical bar for large vertical bar A vertical bar. Third, we show that under mild conditions on f, vertical bar f(A, A)vertical bar is substantially larger than vertical bar A vertical bar, provided that vertical bar A vertical bar is large. We also present a conjecture on what the general sum-product result should be.