On the multiplicative order of the roots of $b{X}^{q^r+1}-a{X}^{q^r}+dX-c$

Abstract

In this paper, we find a lower bound for the order of the group $⟨\theta +\alpha ⟩\subset {\overline{{\mathbb{F}}_q}}^{\ast }$, where $\alpha \in {\mathbb{F}}_q$, $\theta$ is a generic root of the polynomial $F_{A,r}(X)=b{X}^{q^r+1}-a{X}^{q^r}+dX-c\in {\mathbb{F}}_q[X]$ and $ad-bc\ne 0$.

12 pages