Rational Points on Certain Hyperelliptic Curves over Finite Fields

Maciej Ulas. "Rational Points on Certain Hyperelliptic Curves over Finite Fields." Bulletin of the Polish Academy of Sciences. Mathematics 55.2 (2007): 97-104. <http://eudml.org/doc/280660>.

@article{MaciejUlas2007,
abstract = {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 $S_\{k\}^\{i\}: u² = ∏_\{j=1\}^\{k\} g_\{i\}(x_\{j\})$, 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 $C_\{i\}:y² = g_\{i\}(x)$, (i=1,2) defined over a finite field, in polynomial time.},
author = {Maciej Ulas},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
language = {eng},
number = {2},
pages = {97-104},
title = {Rational Points on Certain Hyperelliptic Curves over Finite Fields},
url = {http://eudml.org/doc/280660},
volume = {55},
year = {2007},
}

TY - JOUR
AU - Maciej Ulas
TI - Rational Points on Certain Hyperelliptic Curves over Finite Fields
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2007
VL - 55
IS - 2
SP - 97
EP - 104
AB - 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 $S_{k}^{i}: u² = ∏_{j=1}^{k} g_{i}(x_{j})$, 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 $C_{i}:y² = g_{i}(x)$, (i=1,2) defined over a finite field, in polynomial time.
LA - eng
UR - http://eudml.org/doc/280660
ER -