Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4

Mouhamadou Diaby Gassama; Oumar Sall

doi:doi:10.11648/j.ajam.20221004.15

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Algebraic Points of Degree at Most 3 on the Affine Equation Curve y¹¹=x⁴(x-1)⁴

Mouhamadou Diaby Gassama, Oumar Sall

Published in American Journal of Applied Mathematics (Volume 10, Issue 4)

Received: 29 May 2022 Accepted: 13 July 2022 Published: 18 August 2022

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Abstract

The quotients of Fermat curves C_r,s(p) are studied by Oumar SALL. Among these studies are the cases C_r,s(11) for r = s = 1. Mamina COLY and Oumar SALL have explicitly determined the algebraic points of degree at most 3 on Q for the cases C_r,s(11) for r = s = 2. Our work focuses on determining explicitly the algebraic points of degree at most 3 on Q on the curve C_4,4(11) which is a special case of Fermat quotient curves. Our study concerns the cases C_r,s(11) for r = s = 4. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J_4,4(11)(Q) is an essential condition. So to determine the algebraic points on the curve C_4,4(11) we need a finiteness of the Mordeill-Weill group of rational points of the Jacobien J_4,4(11)(Q). The Mordell-Weil group J_4,4(11)(Q) of rational points of the Jacobien is finite according to Faddev. Our note is in this framework. Our essential tools in this note are the Mordell-Weil group J_4,4(11)(Q) of the Jacobien of C_4,4(11) the Abel-Jacobi theorem and the study of linear systems on the curve C_4,4(11). The result obtained concerns some quotients of Fermat curves. Indeed, the curve C_4,4(11) which is the subject of our study, the set of algebraic points of degree at most 3 on Q has been determined in an explicit way, to achieve this we have determined the quadratic points on the curve C_4,4(11) on Q and the cubic points on the curve C_4,4(11) on Q.

Published in	American Journal of Applied Mathematics (Volume 10, Issue 4)
DOI	10.11648/j.ajam.20221004.15
Page(s)	160-175
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Planes Curves, Degree of Algebraic Points, Jacobien

References

[1]	Arbarello E., Cornalba M., Griffiths P., Harris J., Geometry of Algebraic Curves 1, Grundlehren der Math. Wiss. 267, Springer, New York, 1985.
[2]	C. M. Coly, O. Sall, points algébriques de degré au-plus 3 sur la courbe d’équation affine y¹¹=x²(x-1)², Annales Mathématiques Africaines Volume 8 (2020) pp. 27-32.
[3]	O. Debarre, R. Fahlaoui, Abelian varieties and curves in W (C) and points of bounded degree on algebraic curves, Compositio Math. 88 (1993) 235-249.
[4]	O. Debarre, M. Klassen, Points of low degree on smooth plane curves, J. Reine Angew. Math. 446 (1994) 81-87.
[5]	D. Faddeev, On the divisor class groups of some algebraic curves, Dokl. Akad. Nauk SSSR 136 (1961) 296-298. English translation: Soviet Math. Dokl. 2 (1) (1961) 67- 69.
[6]	B. Gross, D. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve, Invent. Math. 44 (1978) 201-224.
[7]	M. Klassen, P. Tzermias, Algebraic points of low degree on the Fermat quintic, Acta Arith. 82 (4) (1997) 393-401.
[8]	O. Sall, Points algébriques de petit degré sur les courbes de Fermat, C. R. Acad. Sci. Paris Sér. I 330 (2000) 67-70.
[9]	O. Sall, Points cubiques sur la quartique de Klein, C. R. Acad. Sci. Paris Sér. I 333 (2001) 931-934.
[10]	O. Sall, algebraic points on some Fermat curves and some quotients of Fermat curves: Progress, African Journal Of Mathematical Physics Volume 8 (2010) 79-83.
[11]	O. Sall, M. Fall, points algébriques de petits degrés sur les courbes d’équations affines y³ⁿ= x(x−1)(x−2)(x−3), Annales Mathématiques Africaines (2015).
[12]	J. P. Serre, Lecture of Mordell weil theorem. the translated from the french and edited by martin brows from notes by michel Waldschmidt. Aspects of mathematics, E15 (1989).
[13]	J. P. Serre, Représentation des groupes finis, Hermann, Paris, 1967.
[14]	P.Tzermias, AlgebraicpointsoflowdegreeontheFermat curve of degree seven, Manuscriptc Math. 97 (4) (1998) 483-488.
[15]	P. Tzermias, Torsion parts of Mordell-Weil groups of Fermat Jacobians, Internat. Math. Res. Notices 7 (1998) 359-369.

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Mouhamadou Diaby Gassama, Oumar Sall. (2022). Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4. American Journal of Applied Mathematics, 10(4), 160-175. https://doi.org/10.11648/j.ajam.20221004.15

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ACS Style

Mouhamadou Diaby Gassama; Oumar Sall. Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4. Am. J. Appl. Math. 2022, 10(4), 160-175. doi: 10.11648/j.ajam.20221004.15

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AMA Style

Mouhamadou Diaby Gassama, Oumar Sall. Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4. Am J Appl Math. 2022;10(4):160-175. doi: 10.11648/j.ajam.20221004.15

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@article{10.11648/j.ajam.20221004.15,
  author = {Mouhamadou Diaby Gassama and Oumar Sall},
  title = {Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4},
  journal = {American Journal of Applied Mathematics},
  volume = {10},
  number = {4},
  pages = {160-175},
  doi = {10.11648/j.ajam.20221004.15},
  url = {https://doi.org/10.11648/j.ajam.20221004.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221004.15},
  abstract = {The quotients of Fermat curves Cr,s(p) are studied by Oumar SALL. Among these studies are the cases Cr,s(11) for r = s = 1. Mamina COLY and Oumar SALL have explicitly determined the algebraic points of degree at most 3 on Q for the cases Cr,s(11) for r = s = 2. Our work focuses on determining explicitly the algebraic points of degree at most 3 on Q on the curve C4,4(11) which is a special case of Fermat quotient curves. Our study concerns the cases Cr,s(11) for r = s = 4. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J4,4(11)(Q) is an essential condition. So to determine the algebraic points on the curve C4,4(11) we need a finiteness of the Mordeill-Weill group of rational points of the Jacobien J4,4(11)(Q). The Mordell-Weil group J4,4(11)(Q) of rational points of the Jacobien is finite according to Faddev. Our note is in this framework. Our essential tools in this note are the Mordell-Weil group J4,4(11)(Q) of the Jacobien of C4,4(11) the Abel-Jacobi theorem and the study of linear systems on the curve C4,4(11). The result obtained concerns some quotients of Fermat curves. Indeed, the curve C4,4(11) which is the subject of our study, the set of algebraic points of degree at most 3 on Q has been determined in an explicit way, to achieve this we have determined the quadratic points on the curve C4,4(11) on Q and the cubic points on the curve C4,4(11) on Q.},
 year = {2022}
}

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TY  - JOUR
T1  - Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4
AU  - Mouhamadou Diaby Gassama
AU  - Oumar Sall
Y1  - 2022/08/18
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221004.15
DO  - 10.11648/j.ajam.20221004.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 160
EP  - 175
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20221004.15
AB  - The quotients of Fermat curves Cr,s(p) are studied by Oumar SALL. Among these studies are the cases Cr,s(11) for r = s = 1. Mamina COLY and Oumar SALL have explicitly determined the algebraic points of degree at most 3 on Q for the cases Cr,s(11) for r = s = 2. Our work focuses on determining explicitly the algebraic points of degree at most 3 on Q on the curve C4,4(11) which is a special case of Fermat quotient curves. Our study concerns the cases Cr,s(11) for r = s = 4. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J4,4(11)(Q) is an essential condition. So to determine the algebraic points on the curve C4,4(11) we need a finiteness of the Mordeill-Weill group of rational points of the Jacobien J4,4(11)(Q). The Mordell-Weil group J4,4(11)(Q) of rational points of the Jacobien is finite according to Faddev. Our note is in this framework. Our essential tools in this note are the Mordell-Weil group J4,4(11)(Q) of the Jacobien of C4,4(11) the Abel-Jacobi theorem and the study of linear systems on the curve C4,4(11). The result obtained concerns some quotients of Fermat curves. Indeed, the curve C4,4(11) which is the subject of our study, the set of algebraic points of degree at most 3 on Q has been determined in an explicit way, to achieve this we have determined the quadratic points on the curve C4,4(11) on Q and the cubic points on the curve C4,4(11) on Q.
VL  - 10
IS  - 4
ER  -

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Author Information

Mouhamadou Diaby Gassama

Mathematics and Applications Laboratory, Faculty of Science and Technology, Assane Seck University in Ziguinchor, Ziguinchor, Senegal
Oumar Sall

Mathematics and Applications Laboratory, Faculty of Science and Technology, Assane Seck University in Ziguinchor, Ziguinchor, Senegal

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APA Style

Mouhamadou Diaby Gassama, Oumar Sall. (2022). Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4. American Journal of Applied Mathematics, 10(4), 160-175. https://doi.org/10.11648/j.ajam.20221004.15

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ACS Style

Mouhamadou Diaby Gassama; Oumar Sall. Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4. Am. J. Appl. Math. 2022, 10(4), 160-175. doi: 10.11648/j.ajam.20221004.15

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AMA Style

Mouhamadou Diaby Gassama, Oumar Sall. Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4. Am J Appl Math. 2022;10(4):160-175. doi: 10.11648/j.ajam.20221004.15

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@article{10.11648/j.ajam.20221004.15,
  author = {Mouhamadou Diaby Gassama and Oumar Sall},
  title = {Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4},
  journal = {American Journal of Applied Mathematics},
  volume = {10},
  number = {4},
  pages = {160-175},
  doi = {10.11648/j.ajam.20221004.15},
  url = {https://doi.org/10.11648/j.ajam.20221004.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221004.15},
  abstract = {The quotients of Fermat curves Cr,s(p) are studied by Oumar SALL. Among these studies are the cases Cr,s(11) for r = s = 1. Mamina COLY and Oumar SALL have explicitly determined the algebraic points of degree at most 3 on Q for the cases Cr,s(11) for r = s = 2. Our work focuses on determining explicitly the algebraic points of degree at most 3 on Q on the curve C4,4(11) which is a special case of Fermat quotient curves. Our study concerns the cases Cr,s(11) for r = s = 4. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J4,4(11)(Q) is an essential condition. So to determine the algebraic points on the curve C4,4(11) we need a finiteness of the Mordeill-Weill group of rational points of the Jacobien J4,4(11)(Q). The Mordell-Weil group J4,4(11)(Q) of rational points of the Jacobien is finite according to Faddev. Our note is in this framework. Our essential tools in this note are the Mordell-Weil group J4,4(11)(Q) of the Jacobien of C4,4(11) the Abel-Jacobi theorem and the study of linear systems on the curve C4,4(11). The result obtained concerns some quotients of Fermat curves. Indeed, the curve C4,4(11) which is the subject of our study, the set of algebraic points of degree at most 3 on Q has been determined in an explicit way, to achieve this we have determined the quadratic points on the curve C4,4(11) on Q and the cubic points on the curve C4,4(11) on Q.},
 year = {2022}
}

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TY  - JOUR
T1  - Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4
AU  - Mouhamadou Diaby Gassama
AU  - Oumar Sall
Y1  - 2022/08/18
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221004.15
DO  - 10.11648/j.ajam.20221004.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 160
EP  - 175
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20221004.15
AB  - The quotients of Fermat curves Cr,s(p) are studied by Oumar SALL. Among these studies are the cases Cr,s(11) for r = s = 1. Mamina COLY and Oumar SALL have explicitly determined the algebraic points of degree at most 3 on Q for the cases Cr,s(11) for r = s = 2. Our work focuses on determining explicitly the algebraic points of degree at most 3 on Q on the curve C4,4(11) which is a special case of Fermat quotient curves. Our study concerns the cases Cr,s(11) for r = s = 4. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J4,4(11)(Q) is an essential condition. So to determine the algebraic points on the curve C4,4(11) we need a finiteness of the Mordeill-Weill group of rational points of the Jacobien J4,4(11)(Q). The Mordell-Weil group J4,4(11)(Q) of rational points of the Jacobien is finite according to Faddev. Our note is in this framework. Our essential tools in this note are the Mordell-Weil group J4,4(11)(Q) of the Jacobien of C4,4(11) the Abel-Jacobi theorem and the study of linear systems on the curve C4,4(11). The result obtained concerns some quotients of Fermat curves. Indeed, the curve C4,4(11) which is the subject of our study, the set of algebraic points of degree at most 3 on Q has been determined in an explicit way, to achieve this we have determined the quadratic points on the curve C4,4(11) on Q and the cubic points on the curve C4,4(11) on Q.
VL  - 10
IS  - 4
ER  -

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