Estimates for exponential sums with a large automorphism group

2010/10/01 by Rojas-León, Antonio
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1010.0120

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 \AAA¹_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.

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• Ax's theorem with an additive character
• On exponential sums
• Index bounds for character sums with polynomials over finite fields
• On exponential sums over multiplicative subgroups of medium size
• Exponential sum estimates over prime fields