Normal points on Artin–Schreier curves over finite fields
In 2022, S. D. Cohen and the two authors introduced and studied the concept of $(r, n)$-freeness on finite cyclic groups $G$ for suitable integers $r$, $n$, which is an arithmetic way of capturing elements of special forms that lie in the subgroups of $G$. Combining this machinery with some characte...
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Académie des sciences
2025-06-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.740/ |
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| author | Kapetanakis, Giorgos Reis, Lucas |
| author_facet | Kapetanakis, Giorgos Reis, Lucas |
| author_sort | Kapetanakis, Giorgos |
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| description | In 2022, S. D. Cohen and the two authors introduced and studied the concept of $(r, n)$-freeness on finite cyclic groups $G$ for suitable integers $r$, $n$, which is an arithmetic way of capturing elements of special forms that lie in the subgroups of $G$. Combining this machinery with some character sum techniques, they explored the existence of points $(x_0, y_0)$ on affine curves $y^n=f(x)$ defined over a finite field $\mathbb{F}$ whose coordinates are generators of the multiplicative cyclic group $\mathbb{F}^*$. In this paper we develop the natural additive counterpart of this work for finite fields. Namely, any finite extension $\mathbb{E}$ of a finite field $\mathbb{F}$ with $Q$ elements is a cyclic $\mathbb{F}[x]$-module induced by the Frobenius automorphism $\alpha \mapsto \alpha ^{Q}$, and any generator of this module is said to be a normal element over $\mathbb{F}$. We introduce and study the concept of $(f, g)$-freeness on this module structure for suitable polynomials $f, g\in \mathbb{F}[x]$. As a main application of the machinery developed in this paper, we study the existence of $\mathbb{F}_{p^n}$-rational points in the Artin–Schreier curve $\mathfrak{A}_f : y^p-y=f(x)$ whose coordinates are normal over the prime field $\mathbb{F}_p$ and establish concrete results. |
| format | Article |
| id | doaj-art-fd492989d58b438baac0b46ff4fd5655 |
| institution | Matheson Library |
| issn | 1778-3569 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | Académie des sciences |
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| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-fd492989d58b438baac0b46ff4fd56552025-08-01T07:26:11ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692025-06-01363G654155410.5802/crmath.74010.5802/crmath.740Normal points on Artin–Schreier curves over finite fieldsKapetanakis, Giorgos0https://orcid.org/0000-0002-3488-038XReis, Lucas1https://orcid.org/0000-0002-6224-9712Department of Mathematics, University of Thessaly, 3rd km Old National Road Lamia–Athens, 35100, Lamia, GreeceDepartamento de Matemática, Universidade Federal de Minas Gerais, UFMG, Belo Horizonte MG, BrazilIn 2022, S. D. Cohen and the two authors introduced and studied the concept of $(r, n)$-freeness on finite cyclic groups $G$ for suitable integers $r$, $n$, which is an arithmetic way of capturing elements of special forms that lie in the subgroups of $G$. Combining this machinery with some character sum techniques, they explored the existence of points $(x_0, y_0)$ on affine curves $y^n=f(x)$ defined over a finite field $\mathbb{F}$ whose coordinates are generators of the multiplicative cyclic group $\mathbb{F}^*$. In this paper we develop the natural additive counterpart of this work for finite fields. Namely, any finite extension $\mathbb{E}$ of a finite field $\mathbb{F}$ with $Q$ elements is a cyclic $\mathbb{F}[x]$-module induced by the Frobenius automorphism $\alpha \mapsto \alpha ^{Q}$, and any generator of this module is said to be a normal element over $\mathbb{F}$. We introduce and study the concept of $(f, g)$-freeness on this module structure for suitable polynomials $f, g\in \mathbb{F}[x]$. As a main application of the machinery developed in this paper, we study the existence of $\mathbb{F}_{p^n}$-rational points in the Artin–Schreier curve $\mathfrak{A}_f : y^p-y=f(x)$ whose coordinates are normal over the prime field $\mathbb{F}_p$ and establish concrete results.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.740/Finite fieldscharacter sumsnormal elementsfree elementsArtin–Schreier curves |
| spellingShingle | Kapetanakis, Giorgos Reis, Lucas Normal points on Artin–Schreier curves over finite fields Comptes Rendus. Mathématique Finite fields character sums normal elements free elements Artin–Schreier curves |
| title | Normal points on Artin–Schreier curves over finite fields |
| title_full | Normal points on Artin–Schreier curves over finite fields |
| title_fullStr | Normal points on Artin–Schreier curves over finite fields |
| title_full_unstemmed | Normal points on Artin–Schreier curves over finite fields |
| title_short | Normal points on Artin–Schreier curves over finite fields |
| title_sort | normal points on artin schreier curves over finite fields |
| topic | Finite fields character sums normal elements free elements Artin–Schreier curves |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.740/ |
| work_keys_str_mv | AT kapetanakisgiorgos normalpointsonartinschreiercurvesoverfinitefields AT reislucas normalpointsonartinschreiercurvesoverfinitefields |