Estimates for exponential sums with a large automorphism group

Abstract

We prove some improvements of the classical Weil bound for one variable additive and multiplicative character sums associated to a polynomial over a finite field k = Fq for two classes of polynomials which are invariant under a large abelian group of automorphisms of the affine line A 1 k : those invariant under translation by elements of k and those invariant under homotheties with ratios in a large subgroup of the multiplicative group of k. In both cases, we are able to improve the bound by a factor of √ q over an extension of k of cardinality sufficiently large compared to the degree of f .Partially supported by P08-FQM-03894 (Junta de Andalucía), MTM2007-66929 and FEDER.

Finite Noncommutative Geometries Related to $\mathbb {F}_{p}[x]$

“…We start with F p [x] ψ = F p [g 1 ]. This is known from Artin-Schreier theory but for completeness we include an elementary proof from [24]. If f (x+1) = f (x) and f has degree less than p then the proof of Proposition 3.1 with µ = 1 applies and allows us to conclude that f (x) is a constant.…”