Primitive pairs of rational functions with prescribed traces over finite fields

TL;DR

This paper establishes conditions for the existence of elements in finite fields that form primitive pairs with prescribed traces, extending understanding of rational functions and primitive elements in finite field theory.

Contribution

It provides a sufficient condition for primitive pairs with prescribed traces over finite fields and explicitly characterizes exceptions for certain degrees and field sizes.

Findings

Existence of elements forming primitive pairs with prescribed traces under certain conditions.

Explicit characterization of exceptional pairs for degree sum 3 and large field extensions.

Most finite fields admit such elements, with finitely many exceptions.

Abstract

Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of a non-zero element $\epsilon \in \mathbb{F}_{q^m}$, such that $(f(\epsilon), g(\epsilon))$ is a primitive pair in $\mathbb{F}_{q^m}$ with two prescribed traces, $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a$ and $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon^{-1})=b$, where $f(x), g(x) \in \mathbb{F}_{q^m}(x)$ are rational functions with some restrictions and $a, b \in \mathbb{F}_q$. Also, we show that there exists an element $\epsilon \in \mathbb{F}_{q^m}$ satisfying our desired properties in all but finitely many fields $\mathbb{F}_{q^m}$ over $\mathbb{F}_q$. We also calculate possible exceptional pairs explicitly for $m\geq 9$, when degree sums of both the rational functions are taken to be 3.