José Alves Oliveira

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• Phd
• Professor at Federal University of Lavras

Let Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document} be a finite field with q elements and let n be a positive integer. In this paper, we...

In this paper, we study the number of $\mathbb F_{q^n}$-rational points on the affine curve $\mathcal{X}_{d,a,b}$ given by the equation $$ y^d=ax\text{Tr}(x)+b,$$ where $\text{Tr}$ denote the trace function from $\mathbb F_{q^n}$ to $\mathbb F_{q}$ and $d$ is a positive integer. In particular, we present bounds for the number of $\mathbb F_{q}$-rat...

Let Fq denote the finite field with q elements. In this paper we use the relation between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements a∈Fq for which the binomial xn(xq−13+a) is a permutation polynomial. In order to do this, we employ results on the Cartin-Manin operator and the Riemann...

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $n$ be a positive integer. In this paper, we study the digraph associated to the map $x\mapsto x^n h(x^{\frac{q-1}{m}})$, where $h(x)\in\mathbb{F}_q[x].$ We completely determine the associated functional graph of maps that satisfy a certain condition of regularity. In particular, we pro...

Let Fq be a finite field with q=pn elements. In this paper, we study the number of solutions of equations of the form a1x1d1+…+asxsds=b with xi∈Fpti, where ai,b∈Fq and ti|n for all i=1,…,s. In our main results, we employ results on quadratic forms to give an explicit formula for the number of solutions of diagonal equations with restricted solution...

Let F q \mathbb {F}_q be a finite field with q = p n q=p^n elements. In this paper, we study the number of F q \mathbb {F}_q -rational points on the affine hypersurface X \mathcal X given by a 1 x 1 d 1 + ⋯ + a s x s d s = b a_1 x_1^{d_1}+\dots +a_s x_s^{d_s}=b , where b ∈ F q ∗ b\in \mathbb {F}_q^* . A classic well-known result of Weil yields a bo...

Let Fq denote the finite field with q=pλ elements. Maximum Rank metric codes (MRD for short) are subsets of Mm×n(Fq) whose number of elements attains the Singleton-like bound. The first MRD codes known were found by Delsarte (1978) and Gabidulin (1985). Sheekey (2016) presented a new class of MRD codes over Fq called twisted Gabidulin codes and als...

Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ with $x_i\in\mathbb{F}_{p^{t_i}}$, where $b\in\mathbb{F}_q$ and $t_i|n$ for all $i=1,\dots,s$. In our main results, we employ results on quadratic forms to give an explicit formul...

Let Fq denote the finite field with q elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over Fq in terms of the number of rational points on elliptic curves. In the case where q is a prime number, we give a way to calculate these numbers. As a consequence of these results, we characteri...

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. The Carlitz rank of a permutation polynomial is an important measure of complexity of a polynomial. In this paper we find a sharp lower bound for the weight of any permutation polynomial with Carlitz rank 2, improving the bound found by G\'omez-P\'erez, Ostafe and Topuzo\u{g}lu in that c...

Let $\mathbb{F}_q$ be a finite field with $q=p^t$ elements. In this paper, we study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ over $\mathbb{F}_q$. A classic well-konwn result from Weil yields a bound for such number of solutions. In our main result we give an explicit formula for the number of solutions...

Let $\mathbb K$ be a perfect field of characterstic $p\ge 0$ and let $R\in \mathbb K(x)$ be a rational function. This paper studies the number $\Delta_{\alpha, R}(n)$ of distinct solutions of $R^{(n)}(x)=\alpha$ over the algebraic closure $\overline{\mathbb K}$ of $\mathbb K$, where $\alpha\in \overline{\mathbb K}$ and $R^{(n)}$ is the $n$-fold com...

Let $\mathbb{F}_q$ denote the finite field with $q=p^\lambda$ elements. Maximum Rank-metric codes (MRD for short) are subsets of $M_{m\times n}(\mathbb{F}_q)$ whose number of elements attains the Singleton-like bound. The first MRD codes known was found by Delsarte (1978) and Gabidulin (1985). Sheekey (2016) presented a new class of MRD codes over...

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic curves. In the case where $q$ is a prime number, we give a way to calculate these numbers. As a consequence of...

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. The Carlitz rank of a permutation polynomial is a important measure of complexity of the polynomial. In this paper we find the sharp lower bound for the weight of any permutation polynomial with Carlitz rank $2$, improving the bound found by G\'omez-P\'erez, Ostafe and Topuzo\u{g}lu in t...

Let $\mathbb F_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in \mathbb F_q$ for which the binomial $x^n(x^{(q-1)/r} + a)$ is a permutation polynomial in the cases $r = 2$ and $r = 3$.

Neste trabalho, apresentaremos alguns tópicos atuais envolvendo polinômios de permutação sobre corpos finitos. Em especial, exibiremos a teoria necessária para contar o número de binômios de permutação das formas $x^n(x^{\frac{q-1}{2}}+a)$ e $x^n(x^{\frac{q-1}{3}}+a)$. Apresentaremos também o conceito de posto de Carlitz para polinômios de permutaç...