arXiv subjects

José Gustavo Coelho

Publications and source records attributed to José Gustavo Coelho.

2 recordsLinked to original sources

arXiv2025

Solutions of full equations related to diagonal equations

Let $p$ be a prime number, $m$ be an even positive integer, and $\mathbb{F}_q$ be a finite field with $q = p^m$ elements. In this paper, we compute the number of solutions with all coordinates in $\mathbb{F}_q^*$ for diagonal equations of the form $$a_1 x_1^{d} + \dots + a_s x_s^{d} = b, \quad a_i \in \mathbb{F}_q^*, \, b \in \mathbb{F}_q,$$ when the coefficients and exponents satisfy specific arithmetic conditions that facilitate the computation through pure Gauss sums. We then apply this result to determine the number of solutions for equations of the form $$a_1 x_1^{d_{1,1}} \cdots x_n^{d_{n,1}} + \dots + a_s x_1^{d_{1,s}}\cdots x_n^{d_{n,s}} = b,$$ where all exponents are positive, and the equation is related in a particular way to diagonal equations with the aforementioned characteristics.

math.NT↗

arXiv2023

Counting roots of fully triangular polynomials over finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements, $f \in \mathbb{F}_q[x_1, \dots, x_n]$ a polynomial in $n$ variables and let us denote by $N(f)$ the number of roots of $f$ in $\mathbb{F}_q^n$. %Many authors, such as Wei Cao and Kung Jiang have used augmented degree matrices to determine $N(f)$ for different families of polynomials. In this paper we consider the family of fully triangular polynomials, i.e., polynomials of the form \begin{equation*} f(x_1, \dots, x_n) = a_1 x_1^{d_{1,1}} + a_2 x_1^{d_{1,2}} x_2^{d_{2,2}} + \dots + a_n x_1^{d_{1,n}}\cdots x_n^{d_{n,n}} - b, \end{equation*} where $d_{i,j} > 0$ for all $1 \le i \le j \le n$. For these polynomials, we obtain explicit formulas for $N(f)$ when the augmented degree matrix of $f$ is row-equivalent to the augmented degree matrix of a linear polynomial or a quadratic diagonal polynomial.

math.NT↗