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Normal points on Artin-Schreier curves over finite fields

Giorgos Kapetanakis, Lucas Reis

TL;DR

This work develops an additive analogue of freeness in finite fields by viewing $\\mathbb{F}_{q^n}$ as an $\\mathbb{F}_q[x]$-module via the Frobenius map and introducing $(f,g)$-free elements and normal elements. It builds a framework from polynomial arithmetic functions, the $\\mathbb{F}_q$-order, and additive characters to express freeness through character sums and to control counts of elements with prescribed freeness. The main application targets Artin-Schreier curves $\\mathfrak A_f: y^p-y=f(x)$, establishing criteria for the existence of $\\mathbb{F}_q$-rational points with coordinates normal over $\\mathbb{F}_p$ and giving explicit sieve-based bounds; an explicit list of potential exceptional pairs $(n,p)$ is identified and reduced via a prime sieve. The results provide practical methods for constructing normal points on Artin-Schreier curves, with potential implications for coding theory and cryptography due to the role of normal elements in finite field arithmetic.

Abstract

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.

Normal points on Artin-Schreier curves over finite fields

TL;DR

This work develops an additive analogue of freeness in finite fields by viewing as an -module via the Frobenius map and introducing -free elements and normal elements. It builds a framework from polynomial arithmetic functions, the -order, and additive characters to express freeness through character sums and to control counts of elements with prescribed freeness. The main application targets Artin-Schreier curves , establishing criteria for the existence of -rational points with coordinates normal over and giving explicit sieve-based bounds; an explicit list of potential exceptional pairs is identified and reduced via a prime sieve. The results provide practical methods for constructing normal points on Artin-Schreier curves, with potential implications for coding theory and cryptography due to the role of normal elements in finite field arithmetic.

Abstract

In 2022, S.D. Cohen and the two authors introduced and studied the concept of -freeness on finite cyclic groups for suitable integers , which is an arithmetic way of capturing elements of special forms that lie in the subgroups of . Combining this machinery with some character sum techniques, they explored the existence of points on affine curves defined over a finite field whose coordinates are generators of the multiplicative cyclic group . In this paper we develop the natural additive counterpart of this work for finite fields. Namely, any finite extension of a finite field with elements is a cyclic -module induced by the Frobenius automorphism , and any generator of this module is said to be a normal element over . We introduce and study the concept of -freeness on this module structure for suitable polynomials . As a main application of the machinery developed in this paper, we study the existence of -rational points in the Artin-Schreier curve whose coordinates are normal over the prime field and establish concrete results.

Paper Structure

This paper contains 11 sections, 18 theorems, 34 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preparation
  3. Some arithmetic functions over $\mathbb{F}_q[x]$
  4. The $\mathbb{F}_q$-order of an element in $\overline{\mathbb{F}}_q$
  5. Additive characters
  6. Introducing $(f, g)$-freeness
  7. On $(f, g)$-freeness through polynomial values
  8. Normal points on Artin-Schreier curves
  9. Proof of Theorem \ref{['thm:as']}
  10. Acknowledgments
  11. Disclosure statement

Key Result

Theorem 1.1

Let $q=p^n$ be a prime power, where $n\ge 5$ and let $f\in \mathbb{F}_q[x]$ be a polynomial that is not of the form $ax^p+bx+c$ with $a, b, c\in \mathbb{F}_q$ satisfying $1<\deg(f)\le p+1$. Then there exists a point $(x_0, y_0)\in \mathbb{F}_q\times \mathbb{F}_q$ in the Artin-Schreier curve $\mathfr

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 27 more