Mordell type exponential sum estimates in fields of prime order

Abstract

We establish a Mordell type exponential sum estimate (see Mordell [Q. J. Math. 3 (1932) 161–162]) for ‘sparse’ polynomials f ( x ) = ∑ i = 1 r a i x k i , ( a i , p ) = 1 , p prime, under essentially optimal conditions on the exponents 1 ⩽ k i < p − 1 . The method is based on sum–product estimates in finite fields F p and their Cartesian products. We also obtain estimates on incomplete sums of the form ∑ s = 1 t e p ( ∑ i = 1 r a i θ i s ) for t > p ɛ , under appropriate conditions on the θ i ∈ F p * . To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 339 (2004).