Publishing house / Journals and Serials / Bulletin of the Polish Academy of Sciences. Mathematics / All issues

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 $$ S_{k}^{i}:\quad u^2=\prod_{j=1}^{k}g_{i}(x_{j}),\quad\ i=1,2, $$ contains a rational curve. Using the above and van de Woestijne's recent results we show how to 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.