On the multiplicative order of the roots of $bX^{q^r+1}-aX^{q^r}+dX-c$

TL;DR

This paper establishes a lower bound on the multiplicative order of roots of a specific polynomial over finite fields, contributing to understanding their algebraic structure and properties.

Contribution

It provides a new lower bound for the order of roots of a class of polynomials over finite fields, extending previous results in algebraic number theory.

Findings

Lower bound for the order of roots of the polynomial

Conditions under which the bound applies

Insights into the structure of roots in finite fields

Abstract

In this paper, we find a lower bound for the order of the group $\langle \theta+\alpha\rangle \subset \overline{\mathbb F_{q}}^*$, where $\alpha\in \mathbb F_{q}$, $\theta$ is a generic root of the polynomial $F_{A,r}(X)=bX^{q^r+1}-aX^{q^r}+dX-c\in \mathbb F_{q}[X]$ and $ad-bc\ne0$.