Rational Points on Certain Hyperelliptic Curves over Finite Fields

Abstract

Let $K$ be a field, $a,b\in K$ and $ab\neq 0$. Consider the polynomials $g_{1}(x)=x^n+ax+b$, $g_{2}(x)=x^n+ax^2+bx$, where $n$ is a fixed positive integer. We show that for each $k\geq 2$ the hypersurface given by the equation $$ S_{k}^{i}:\quad u^2=\prod