Existence of Special Types Primitive Pairs in Finite Fields Avoiding Affine Hyperplanes

Abstract

Let $\Fm$ be finite fields of order $q^m$, where $m\ge 2$ and $q$, a prime power. Given $\F$-affine hyperplanes $A_1,\dots ,A_m$ of $\Fm$ in general position, we study the existence of primitive element $\alpha$ of $\Fm$, such that $f(\alpha )$ is also primitive, where $ax^2+bx+c\in \Fm[x]$ ($a\ne 0$ and $b^2\ne 4ac$) in $\Fm$ and the primitive pair $(\alpha ,f(\alpha ))$ avoids each $A_i$. We establish results for fields of higher order.

12 pages