Mordell type exponential sum estimates in fields of prime order

We establish a Mordell type exponential sum estimate (see Mordell Q. J. Math. 3 (1932) 161–162]) for ‘sparse’ polynomials $f(x)={\sum }_{i=1}^r{a}_i{x}^{k_i}\text{,}(a_i\text{,}p)=1\text{,}p$ prime, under essentially optimal conditions on the exponents $1?k_i<p?1$. The method is based on sum–product estimates in finite fields ${\mathbb{F}}_p$ and their Cartesian products. We also obtain estimates on incomplete sums of the form ${\sum }_{s=1}^t{e}_p({\sum }_{i=1}^r{a}_i{\theta }_i^s)$ for $t>p^{?}$, under appropriate conditions on the ${\theta }_i\in {\mathbb{F}}_p^{*}$. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 339 (2004).