On families of additive exponential sums

where f is a polynomial of degree e in n variables with coefficients in a finite field F of characteristic p, E is an overfield of F, and ψ is an additive character of F. It is interesting to ask what happens with sums S(E, f) when one varies E, or when one varies f . In order to talk precisely about the latter, we denote by P(e, n) the scheme parameterizing all polynomials in n variables of degree ≤ e with coefficients in the algebraic closure F. First, suppose one fixes the polynomial f , and varies E. Then Deligne showed that if f lies in a Zariski dense subset of the parameter space P(e, n) the sums S(E, f) can be evaluated