Existence of primitive pairs with two prescribed traces over finite fields

Abstract

Given $F={\mathbb{F}}_{p^t}$, a field with $p^t$ elements, where $p$ is a prime power, $t\ge 7$, $n$ are positive integers and $f=f_1/f_2$ is a rational function, where $f_1,f_2$ are relatively prime, irreducible polynomials with $deg(f_1)+deg(f_2)=n$ in $F[x]$. We construct a sufficient condition on $(p,t)$ which guarantees primitive pairing $(\epsilon ,f(\epsilon ))$ exists in $F$ such that $T{r}_{{\mathbb{F}}_{p^t}/{\mathbb{F}}_p}(\epsilon )=a$ and $T{r}_{{\mathbb{F}}_{p^t}/{\mathbb{F}}_p}(f(\epsilon ))=b$ for any prescribed $a,b\in {\mathbb{F}}_p$. Further, we demonstrate for any positive integer $n$, such a pair definitely exists for large $t$. The scenario when $n=2$ is handled separately and we verified that such a pair exists for all $(p,t)$ except from possible 71 values of $p$. A result for the case $n=3$ is given as well.