Rational points on certain hyperelliptic curves over finite fields

• Autorzy: Ulas M.
• 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[...], 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 C1 : y2 = gi(x), (i = 1,2) defined over a finite field, in polynomial time.

Słowa kluczowe

EN

hyperelliptic curves; rational points; diophantine equations; finite fields

PL

krzywe hipereliptyczne; równanie diofantyczne

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2007
• Tom: Vol. 55, no 2
• Strony: 97--104
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Ulas M.

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

Bibliografia

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• [2] J. F. Mestre, Rang de courbes elliptiques d'invariant donne, C. R. Acad. Sci. Paris Ser. I Math. 314 (1992), 919-922.
• [3] A. Schinzel and M. Skalba, On equations y2 = xn + k in a finite field, Bull. Polish Acad. Sci. Math. 52 (2004), 223-226.
• [4] R. Schoof, Elliptic curves over finite fields and the computation of square roots modp, Math. Comp. 44 (1985), 483-494.
• [5] A. Shallue and Ch. van de Woestijne, Construction of rational points on elliptic curves over finite fields, in: Algorithmic Number Theory, Lecture Notes in Comput. Sci. 4076, Berlin, 2006, 510-524.
• [6] D. Shanks, Five number-theoretic algorithms, Congr. Numer. 7 (1972), 51-70.
• [7] M. Skalba, Points on elliptic curves over finite fields, Acta Arith. 117 (2005), 293-301.
• [8] Ch. van de Woestijne, Deterministic equation solving over finite fields, PhD thesis,Univ. Leiden, 2006.