The additive exponential sums associated to an integral polynomial

satisfy a property of twisted-multiplicativity. Using this, it is

possible to exploit properties of these sums over finite fields to gain

some understanding of the sums modulo all integers. This involves a

fine interplay of algebraic methods and analytic techniques. The

explain will describe some of these, and explain in particular how to

deduce that the mean value of these exponential sums vanishes for

suitably generic polynomials.

(Joint work with K. Soundararajan)