Counting roots of fully triangular polynomials over finite fields

TL;DR

This paper derives explicit formulas for counting roots of fully triangular polynomials over finite fields, focusing on cases where the augmented degree matrix is row-equivalent to linear or quadratic diagonal matrices.

Contribution

It introduces formulas for the number of roots of fully triangular polynomials over finite fields based on their augmented degree matrices.

Findings

Explicit formulas for roots when augmented degree matrix is linear-equivalent.

Formulas for quadratic diagonal polynomial cases.

Enhanced understanding of polynomial root structures 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.