Exponential Sums, Twisted Multiplicativity, and Moments

Abstract

We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of exponential sums associated to different suitably unrelated and generic polynomials. The proofs combine analytic arguments with the algebraic interpretation of exponential sums and their monodromy groups.

Sidon sets in algebraic geometry

“…2.8.1 and Th. 7.9.6]), the assumption that the critical values of a polynomial, or the set of parameters of a hypergeometric differential equation, form a Sidon set or a symmetric Sidon set in the additive group C lead to computations of certain monodromy groups or differential Galois groups; here, even sets of 3 elements give non-trivial results, and such computations in turn have a number of important implications (see, for instance, [8] and [15] for recent examples).…”