Rational Points on Certain Hyperelliptic Curves over Finite Fields

Maciej Ulas¹

¹ Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Krak/ow, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) no. 2, pp. 97-104

Résumé

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.

Détail

DOI : 10.4064/ba55-2-1

Keywords: field neq consider polynomials where fixed positive integer each geq hypersurface given equation quad prod quad contains rational curve using above van woestijnes recent results construct rational point different point infinity curves defined finite field polynomial time