On Bilinear Exponential and Character Sums with Reciprocals of Polynomials

TL;DR

This paper establishes new bounds for bilinear exponential sums involving reciprocals of polynomials over finite fields, with applications to Kloosterman sums and multiplicative characters, advancing understanding in finite field exponential sum estimates.

Contribution

The paper introduces novel bounds for bilinear sums with reciprocals of polynomials, including linear and Kloosterman fractions, over finite fields, extending previous results.

Findings

New bounds for bilinear sums with linear polynomials

Improved estimates for Kloosterman fractions sums

Enhanced bounds for sums with multiplicative characters

Abstract

We give nontrivial bounds for the bilinear sums $$ \sum_{u = 1}^{U} \sum_{v=1}^V \alpha_u \beta_v \mathbf{\,e}_p(u/f(v)) $$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$ elements, with integers $U$, $V$, a polynomial $f\in {\mathbb F}_p[X] $ and some complex weights $\{\alpha_u\}$, $\{\beta_v\}$. In particular, for $f(X)=aX+b$ we obtain new bounds of bilinear sums with Kloosterman fractions. We also obtain new bounds for similar sums with multiplicative characters of ${\mathbb F}_p$.