Counting roots of fully triangular polynomials over finite fields

Abstract

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.