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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.07884 |
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| _version_ | 1866912265339404288 |
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| author | Kapetanakis, Giorgos Reis, Lucas |
| author_facet | Kapetanakis, Giorgos Reis, Lucas |
| contents | 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 $α\mapsto α^{Q}$, 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 | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07884 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normal points on Artin-Schreier curves over finite fields Kapetanakis, Giorgos Reis, Lucas Number Theory 11T30 (Primary) 11T06, 11T23 (Secondary) 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 $α\mapsto α^{Q}$, 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. |
| title | Normal points on Artin-Schreier curves over finite fields |
| topic | Number Theory 11T30 (Primary) 11T06, 11T23 (Secondary) |
| url | https://arxiv.org/abs/2412.07884 |