Existence of primitive pairs with two prescribed traces over finite fields

Given $F= \mathbb{F}_{p^{t}}$, a field with $p^t$ elements, where $p$ is a prime power, $t\geq 7$, $n$ 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 $(ε, f(ε))$ exists in $F$ such that $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε) = a$ and $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε)) = b$ 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.

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