Let K be a field, a,b ∈ K and ab ≠ 0. Consider the polynomials g₁(x) = xⁿ+ax+b, g₂(x) = xⁿ+ax²+bx, where n is a fixed positive integer. We show that for each k≥ 2 the hypersurface given by the equation , 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 , (i=1,2) defined over a finite field, in polynomial time.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-2-1,
author = {Maciej Ulas},
title = {Rational Points on Certain Hyperelliptic Curves over Finite Fields},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {55},
year = {2007},
pages = {97-104},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-2-1}
}
Maciej Ulas. Rational Points on Certain Hyperelliptic Curves over Finite Fields. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) pp. 97-104. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-2-1/