  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.
