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

TL;DR

This paper investigates the existence of primitive element pairs in finite fields that avoid certain affine hyperplanes, extending results to fields of higher order and focusing on special polynomial forms.

Contribution

It introduces new conditions for primitive pairs in finite fields that avoid affine hyperplanes, particularly for higher order fields and specific quadratic polynomials.

Findings

Primitive pairs avoiding affine hyperplanes exist under certain conditions.

Results extend to finite fields of higher order.

Conditions for quadratic polynomial images of primitive elements are established.

Abstract

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