Purity of exponential sums on An

We give a purity result for two kinds of exponential sums of the type Sigma(x is an element of kn) psi(f(x)), where k is a finite field of characteristic p and psi : k --> (C* is a non-trivial additive character. In the first case, f is an element of k[x(1),...,.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.