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