On the number of solutions of polynomial equation over $\mathbb{F}_p$

Ilya Vyugin, Sergey Makarychev

Published 2015-03-26Version 1

We present an upper bound of the number of solutions $(x,y)$ of a polynomial equation $P(x,y)=0$ over a field $\mathbb{F}_p$, 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 $\mathbb{F}_p^*$. Some applications of this bound to hyperelliptic curves and additive energies are obtained.