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Josimar J. R. Aguirre

Publications and source records attributed to Josimar J. R. Aguirre.

7 recordsLinked to original sources
arXiv2026

Effective existence of primitive k-normal elements via affine spaces

Let $\mathbb{F}_{q^n}$ be the finite field with $q^n$ elements. An element $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if $α$ and its conjugates span an $\mathbb{F}_q$-subspace of dimension $n-k$. It is known that, for $q$ sufficiently large, a primitive $k$-normal element exists if and only if there exists a $k$-normal element free of binomials, but the lower bound on $q$ is not explicit. We count the $k$-normal elements in specific affine subspaces of $\mathbb{F}_{q^n}$ and show that, in some of them, all but at most one element is $k$-normal. Combining this with character sum estimates over affine spaces and a sieve, we obtain an explicit lower bound on $q$ above which a primitive $k$-normal element exists, provided there is a $k$-normal element free of binomials whose $\mathbb{F}_q$-order has a linear factor or is irreducible. As an application, we compute these bounds for $5 \le n \le 8$ and give a complete characterization of the existence of primitive $k$-normal elements in $\mathbb{F}_{q^6}$ over $\mathbb{F}_q$.

math.NT↗
arXiv2023

$r$-primitive $k$-normal elements in arithmetic progressions over finite fields

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. For a positive divisor $r$ of $q^n-1$, the element $α\in \mathbb{F}_{q^n}^*$ is called \textit{$r$-primitive} if its multiplicative order is $(q^n-1)/r$. Also, for a non-negative integer $k$, the element $α\in \mathbb{F}_{q^n}$ is \textit{$k$-normal} over $\mathbb{F}_q$ if $\gcd(αx^{n-1}+ α^q x^{n-2} + \ldots + α^{q^{n-2}}x + α^{q^{n-1}} , x^n-1)$ in $\mathbb{F}_{q^n}[x]$ has degree $k$. In this paper we discuss the existence of elements in arithmetic progressions $\{α, α+β, α+2β, \ldotsα+(m-1)β\} \subset \mathbb{F}_{q^n}$ with $α+(i-1)β$ being $r_i$-primitive and at least one of the elements in the arithmetic progression being $k$-normal over $\mathbb{F}_q$. We obtain asymptotic results for general $k, r_1, \dots, r_m$ and concrete results when $k = r_i = 2$ for $i \in \{1, \dots, m\}$.

math.NT↗
arXiv2022

Pairs of $r$-primitive and $k$-normal elements in finite fields

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements and $r$ be a positive divisor of $q^n-1$. An element $α\in \mathbb{F}_{q^n}^*$ is called $r$-primitive if its multiplicative order is $(q^n-1)/r$. Also, $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_α(x) = αx^{n-1}+ α^q x^{n-2} + \ldots + α^{q^{n-2}}x + α^{q^{n-1}}$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$. These concepts generalize the ideas of primitive and normal elements, respectively. In this paper, we consider non-negative integers $m_1,m_2,k_1,k_2$, positive integers $r_1,r_2$ and rational functions $F(x)=F_1(x)/F_2(x) \in \mathbb{F}_{q^n}(x)$ with $°(F_i) \leq m_i$ for $i\in\{ 1,2\}$ satisfying certain conditions and we present sufficient conditions for the existence of $r_1$-primitive $k_1$-normal elements $α\in \mathbb{F}_{q^n}$ over $\mathbb{F}_q$, such that $F(α)$ is an $r_2$-primitive $k_2$-normal element over $\mathbb{F}_q$. Finally as an example we study the case where $r_1=2$, $r_2=3$, $k_1=2$, $k_2=1$, $m_1=2$ and $m_2=1$, with $n \ge 7$.

math.NT↗
arXiv2022

Number of $k$-normal elements over a finite field

An element $α\in \mathbb{F}_{q^n}$ is a normal element over $\mathbb{F}_q$ if the conjugates $α^{q^i}$, $0 \leq i \leq n-1$, are linearly independent over $\mathbb{F}_q$. Hence a normal basis for $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ is of the form $\{α,α^q, \ldots, α^{q^{n-1}}\}$, where $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$. In 2013, Huczynska, Mullen, Panario and Thomson introduce the concept of k-normal elements, as a generalization of the notion of normal elements. In the last few years, several results have been known about these numbers. In this paper, we give an explicit combinatorial formula for the number of $k$-normal elements in the general case, answering an open problem proposed by Huczynska et al. (2013).

math.CO↗
arXiv2021

About $r$- primitive and $k$-normal elements in finite fields

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of $k$-normal elements: an element $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_α(x)= αx^{n-1}+α^qx^{n-2}+\ldots +α^{q^{n-2}}x+α^{q^{n-1}}$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$, generalizing the concept of normal elements (normal in the usual sense is $0$-normal). In this paper we discuss the existence of $r$-primitive, $k$-normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, where an element $α\in \mathbb{F}_{q^n}^*$ is $r$-primitive if its multiplicative order is $\frac{q^n-1}{r}$. We provide many general results about the existence of this class of elements and we work a numerical example over finite fields of characteristic $11$.

math.NT↗
arXiv2020

Existence of primitive $2$-normal elements in finite fields

An element $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if $\mathcal{B}=\{α, α^q, α^{q^2}, \cdots, α^{q^{n-1}}\}$ forms a basis of $\mathbb{F}_{q^n}$ as a vector space over $\mathbb{F}_q$. It is well known that $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if and only if $g_α(x)=αx^{n-1}+α^q x^{n-2}+ \cdots + α^{q^{n-2}}x+α^{q^{n-1}}$ and $x^n-1$ are relatively prime over $\mathbb{F}_{q^n}$, that is, the degree of their greatest common divisor in $\mathbb{F}_{q^n}[x]$ is $0$. Using this equivalence, the notion of $k$-normal elements was introduced in Huczynska et al. ($2013$): an element $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_α[x]$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$; so an element which is normal in the usual sense is $0$-normal. Huczynska et al. made the question about the pairs $(n,k)$ for which there exist primitive $k$-normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ and they got a partial result for the case $k=1$, and later Reis and Thomson ($2018$) completed this case. The Primitive Normal Basis Theorem solves the case $k=0$. In this paper, we solve completely the case $k=2$ using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of $k$-normal elements in $\mathbb{F}_{q^n}$.

math.NT↗