Research Article

Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve

Volume: 4 Number: 2 August 31, 2021
EN

Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve

Abstract

In this paper, we give a parametrization of algebraic points of degree at most $4$ over $\mathbb{Q}$ on the schaeffer curve $\mathcal{C}$ of affine equation : $ y^{2}=x^{5}+1 $. The result extends our previous result which describes in [5] ( Afr. Mat 29:1151-1157, 2018) the set of algebraic points of degree at most $3$ over $\mathbb{Q}$ on this curve.

Keywords

Degree of algebraic points , Plan curve , Rational points

References

  1. [1] N. Bruin, M. Stoll, The Mordell-Weil sieve : proving the nonexistence of Rational points on curves, LMS J. Comp. Math., 13 (2010), 272 -306.
  2. [2] R. F. Coleman, Effective Chabauty Duke Math. J. 52(3) (1985), 765-770.
  3. [3] E. F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Mathematische Annalen, 310 (1998), 447–471.
  4. [4] M. J. Klassen, E. F. Schaefer, Arithmetic and geometry of the curve x4 = y3 +1, Acta Arithmetica LXXIV.3 (1996) 241-257.
  5. [5] M. Fall, O. Sall, Ponts alg´ebriques de petit degr´e sur la courbe d’´equation affine y2 = x5 +1, Afr. Mat. 29 (2018) 1151-1157.
  6. [6] O. Sall, Points alg´ebriques sur certains quotients de courbes de Fermat, C. R. Acad. Sci. Paris S´er I 336 (2003) 117-120.
  7. [7] S. Siksek, M. Stoll, Partial descent on hyper elliptic curves and the generalized Fermat equation x3 +y4 +z5 = 0, Bulletin of the LMS 44 (2012) 151 -166
APA
Fall, M. (2021). Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve. Journal of Mathematical Sciences and Modelling, 4(2), 51-55. https://doi.org/10.33187/jmsm.931258
AMA
1.Fall M. Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve. Journal of Mathematical Sciences and Modelling. 2021;4(2):51-55. doi:10.33187/jmsm.931258
Chicago
Fall, Moussa. 2021. “Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve”. Journal of Mathematical Sciences and Modelling 4 (2): 51-55. https://doi.org/10.33187/jmsm.931258.
EndNote
Fall M (August 1, 2021) Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve. Journal of Mathematical Sciences and Modelling 4 2 51–55.
IEEE
[1]M. Fall, “Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve”, Journal of Mathematical Sciences and Modelling, vol. 4, no. 2, pp. 51–55, Aug. 2021, doi: 10.33187/jmsm.931258.
ISNAD
Fall, Moussa. “Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve”. Journal of Mathematical Sciences and Modelling 4/2 (August 1, 2021): 51-55. https://doi.org/10.33187/jmsm.931258.
JAMA
1.Fall M. Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve. Journal of Mathematical Sciences and Modelling. 2021;4:51–55.
MLA
Fall, Moussa. “Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve”. Journal of Mathematical Sciences and Modelling, vol. 4, no. 2, Aug. 2021, pp. 51-55, doi:10.33187/jmsm.931258.
Vancouver
1.Moussa Fall. Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve. Journal of Mathematical Sciences and Modelling. 2021 Aug. 1;4(2):51-5. doi:10.33187/jmsm.931258

Cited By

Annales Universitatis Paedagogicae Cracoviensis

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

https://doi.org/10.2478/aupcsm-2023-0003