Given $\mathbb{ F}_{ q^{ n}}$, a field with $q^n$ elements, where $q $ is a prime power and $n$ is positive integer. For $r_1,r_2,m_1,m_2 \in \mathbb{ N}$, $k_1,k_2 \in \mathbb{ N}\cup \{ 0\}$, a rational function $F = \frac{ F_1}{ F_2}$ in $\mathbb{ F}_{ q}[x]$ with deg($F_i$) $\leq m_i$; $i=1,2,$ satisfying some conditions, and $a,b \in \mathbb{ F}_{ q}$, we construct a sufficient condition on $(q,n)$ which guarantees the existence of an $r_1$-primitive, $k_1$-normal element $\epsilon \in \mathbb{ F}_{ q^n}$ such that $F(\epsilon)$ is $r_2$-primitive, $k_2$-normal with $\operatorname{ Tr}_{ \mathbb{ F}_{ q^n}/\mathbb{ F}_q}(\epsilon) = a$ and $\operatorname{ Tr}_{ \mathbb{ F}_{ q^n}/\mathbb{ F}_q}(\epsilon^{ -1}) = b$. For $m_1=10, \; m_2=11,\; r_1 = 3, \; r_2 = 2, \; k_1=2,\;k_2 = 1$, we establish bounds on $q$, for various $n$, to determine the existence of such elements in $\mathbb{ F}_{ q^{ n}}$. Furthermore, we identify all such pairs $(q,n)$ excluding 10 possible values of $(q,n)$, in fields of characteristics 13.
Categories: math.NT