Solutions of polynomial equation over $\mathbb{F}_p$ and new bounds of additive energy

TL;DR

This paper offers a new proof for an existing upper bound on solutions to polynomial equations over finite fields, improves average estimates, and introduces new bounds for additive and polynomial energy.

Contribution

It provides a novel proof of an established bound, enhances average estimates, and establishes new bounds for additive and polynomial energy in finite fields.

Findings

Improved average bounds for solutions of polynomial equations over finite fields.

New bounds for additive energy in finite fields.

Enhanced estimates for polynomial energy.

Abstract

We present a new proof of Corvaja and Zannier's \cite{C-Z} the upper bound of the number of solutions $(x,y)$ of the algebraic equation $P(x,y)=0$ over a field $\mathbb{F}_p$ ($p$ is a prime), in the case, where $x\in g_1G$, $y\in g_2G$, ($g_1G$, $g_2G$ -- are cosets by some subgroup $G$ of a multiplicative group $\mathbb{F}_p^*$). The estimate of Corvaja and Zannier was improved in average, and some applications of it has been obtained. In particular we present the new bounds of additive and polynomial energy.