Primitive pairs of rational functions with prescribed traces over finite fields

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.