El Hadji Sow, Moussa Fall, Oumar Sall
U.F.R. of Science and Technology, Mathematics and Applications Laboratory, Assane SECK University of Ziguinchor, Senegal
Correspondence to: El Hadji Sow, U.F.R. of Science and Technology, Mathematics and Applications Laboratory, Assane SECK University of Ziguinchor, Senegal.
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Copyright © 2022 The Author(s). Published by Scientific & Academic Publishing.
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Abstract
In this work, we determine the set of algebraic points of given degree over on the curve of affine equation This note extends a result of Booker, Sijsling, Sutherland, Voight and Yasak in [1] who gave a description of the set of -rational points i.e the set of points of degree one over on this curve.
Keywords:
Planes curves, Degree of algebraic points, Rationals points, Algebraic extensions, Jacobian
Cite this paper: El Hadji Sow, Moussa Fall, Oumar Sall, Algebraic Points of Given Degree on the Affine Curve , American Journal of Mathematics and Statistics, Vol. 12 No. 1, 2022, pp. 5-8. doi: 10.5923/j.ajms.20221201.02.
1. Introduction
Let be a smooth algebraic curve defined over Let be a numbers field. We note by the set of points of with coordinates in and the set of points of with coordinates in of degree at most over The goal is to determine the set of algebraic points of given degree over on the curve given by the affine equation The Mordell-Weil group of rational points of the Jacobian is a finite set (refer to [1,4]).We denote by: and the point at infinity. In [1] Booker, Sijsling, Sutherland, Voight and Yasak gave a description of the rational points over on this curve. This description is as follows:Proposition: The -rational points on are given by In this note, we give an explicit description of algebraic points of given degree over on the curve .Our main result is given by the following theorem:Theorem: The set of algebraic points of given degree over on the curve isgiven by:With:andwhere and
2. Auxiliary Results
For a divisor on , we note the -vector space of rational functions defined on such that or ; designates the of . In [1, 4] the Mordell-Weil group of is isomorph to and is a hyperelliptic curve of genus Let be two rational functions on defined as follow: The projective equation of We denote by and let's put for Let us designate by the intersection cycle of algebraic curve defined on and .Lemma 1:Consequence of lemma 1: Lemma 2:Proof:- We have since if then the curve is of genus zero (see [1,4]), which is not the case.- Since the genus of is equal to 2, then is a canonical divisor of - For the rest we apply the Riemann-Roch theorem which says that Lemma 3:A -base of is given byProof: It is clear that is free and it remains to show that According to the Riemann-Roch theorem, we have According to the parity of we have the following two cases:Case 1: Suppose that is even and let Thus we have and we have . Then we get . we have Case 2: Suppose that is odd and let Thus we have and Then we get We have Lemma 4:Proof: (See [1,4]).
3. Proof of Theorem
Given The work of Booker, Sijsling, Sutherland, Voight and Yasak in [1] allows us to assume that Note that are the Galois conjugates of Let's work with according to lemma 4 we have So we have Our proof is divided in three cases:Case We have then there exist a function with coefficient in such that then and according to lemma 3 we have For the points we have hence and the relation gives the equation We find a family of points Cases and For we have then there exist a function with coefficient in such that then and according to lemma 3 we have We have For the points we have hence and the relation gives the equation We find a family of points For we have By a similar argument as in case we have Finally, we have the family Cases and For , we have then there exist a function with coefficient in such that then and according to lemma 3 we have The function is of order 2 at point P so we must haveFor the points we have hence and the relation gives the equation We find a family of points For we have By a similar argument as in case we have Finally, we have the family
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