Estimates for exponential sums with a large automorphism group

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 A1 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.f k of cardinality sufficiently large compared to the degree of f. Junta de Andalucía Ministerio de Educación y Ciencia Fondo Europeo de Desarrollo Regional