Purity of exponential sums on $\mathbb{A}^n$

Abstract

We give a purity result for two kinds of exponential sums of the type $\sum_{x\in k^n}\psi(f(x))$, where k is a finite field of characteristic p and $\psi:k\to{\mathbb C}^\star$ is a non-trivial additive character. In the first case, $f\in k[x_1,\dots,x_n]$ is a polynomial of degree divisible by p whose highest-degree homogeneous form defines a non-singular projective hypersurface, and in the second case, f is a polynomial of degree prime to p whose highest-degree homogeneous form defines a projective hypersurface with isolated singularities.