Further improvement on index bounds

TL;DR

This paper improves existing bounds on character sums of polynomials over finite fields, providing tighter estimates and applications to counting solutions of algebraic curves.

Contribution

The authors introduce a new bound that surpasses the Weil and Wan-Wang bounds for character sums, with practical applications in algebraic curve solution counting.

Findings

New bound is tighter than Weil and Wan-Wang bounds

Application to counting solutions of algebraic curves

Examples demonstrate the effectiveness of the new bound

Abstract

In this paper we obtain further improvement of index bounds for character sums of polynomials over finite fields. We present some examples, which show that our new bound is an improved bound compared to both the Weil bound and the index bound given by Wan and Wang. As an application, we count the number of all the solutions of some algebraic curves by using our result.