Exponential sums with sparse polynomials over finite fields

TL;DR

This paper establishes new bounds for exponential sums involving sparse polynomials over finite fields, improving previous results especially for binomials by leveraging gcd-based techniques.

Contribution

It introduces novel bounds for exponential sums with sparse polynomials, notably enhancing results for binomials through gcd-based analysis.

Findings

New bounds for exponential sums with sparse polynomials.

Improved bounds specifically for binomials.

Bounds depend on gcds of exponents and their differences.

Abstract

We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In particular, two new bounds for binomials are obtained, improving previous results in broad ranges of parameters.