Exponential sums and singular hypersurfaces (Q1269493)

The author obtains upper bounds for the absolute values of exponential sums associated to polynomials over finite fields. If \(f\in k[x_1,\dots,x_n]\) and \(\Psi\) is a non-trivial additive character, he puts \(S(\Psi,f) = \sum_{x\in k^n} \Psi(\text{Tr}_{k/{\mathbb F}_p}(f(x)))\). Supposing that \(k\) has \(q\) elements, it is proved that \[ |S(\Psi,\overline{f})|\leq\dim_{\mathbb C} { { {\mathbb C}[x_1,\dots,x_n] } \over { \bigl(\delta f/\delta x_1, \dots, \delta f/\delta x_n\bigr)} } q^{n/2}. \] The proof is based on the reduction to the case \(f=f_d + x_n^{d-1}\), on techniques of \textit{N. Katz} on exponential sums [Sommes exponentielles, Astérisque 79 (1980; Zbl 0469.12007)] and of \textit{M. Raynaud} on cohomology of abelian varieties [in Sémin. Bourbaki 17, Années 1964/65, Exp. No. 286 (1966; Zbl 0204.54301)].