Rational Points on Certain Hyperelliptic Curves over Finite Fields

• Autorzy: Maciej Ulas
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11D25: Cubic and quartic equations
• 11D41: Higher degree equations; Fermat's equation
• 14G15: Finite ground fields

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2007
• Tom: 55
• Numer: 2
• Strony: 97-104

Daty

wydano

2007

Twórcy

autor

Maciej Ulas

• Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

Identyfikatory

DOI

10.4064/ba55-2-1