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 $x\in g_{1}G$, $y\in g_{2}G$, ($g_{1}G$, $g_{2}G$ -- are cosets by some subgroup $G$ of a multiplicative group ${\mathbb{F}}_p^{\ast }$). 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.

11 pages

References in corpus (1)

• Intersections of multiplicative subgroups and Heilbronn's exponential sum

Cited by in corpus (1)

• On tripling constant of multiplicative subgroups