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American Journal of Mathematics and Statistics

p-ISSN: 2162-948X    e-ISSN: 2162-8475

2022;  12(1): 5-8

doi:10.5923/j.ajms.20221201.02

Received: Feb. 16, 2022; Accepted: Mar. 4, 2022; Published: Mar. 15, 2022

### Algebraic Points of Given Degree on the Affine Curve

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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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.

### Article Outline

1. Introduction
2. Auxiliary Results
3. Proof of Theorem

### 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:
and
where
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 by
Proof:
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 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

### References

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