Rational points on certain hyperelliptic curves over finite fields

Abstract

Let $K$ be a field, $a,b\in K$ and $ab\ne 0$. Let us consider the polynomials $g_1(x)=x^n+ax+b,g_2(x)=x^n+a{x}^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $k\ge 2$ the hypersurface given by the equation \begin{equation*} S_{k}^{i}: u^2=\prod_{j=1}^{k}g_{i}(x_{j}),\quad i=1, 2. \end{equation*} contains a rational curve. Using the above and Woestijne's recent results \cite{Woe} we show how one can construct a rational point different from the point at infinity on the curves $C_i:y^2=g_i(x),(i=1,2)$ defined over a finite field, in polynomial time.

Revised version will appear in Bull. Polish Acad. Sci. Math