Primitive normal pairs with prescribed traces over finite fields

<p>Let <span class="math inline">\(q\)</span> be a positive integral power of some prime <span class="math inline">\(p\)</span> and <span class="math inline">\(\mathbb{F}_{q^m}\)</span> be a finite field with <span class="math inline">\(q^m\)</span> elements for some <span class="math inline">\(m \in \mathbb{N}\)</span>. Here we establish a sufficient condition for the existence of primitive normal pairs of the type <span class="math inline">\((\epsilon, f(\epsilon))\)</span> in <span class="math inline">\(\mathbb{F}_{q^m}\)</span> over <span class="math inline">\(\mathbb{F}_{q}\)</span> with two prescribed traces, <span class="math inline">\(\text{Tr}_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\)</span> and <span class="math inline">\(\text{Tr}_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b\)</span>, where <span class="math inline">\(f(x) \in \mathbb{F}_{q^m}(x)\)</span> is a rational function with some restrictions and <span class="math inline">\(a, b \in \mathbb{F}^*_q\)</span>. Furthermore, for <span class="math inline">\(q=5^k\)</span>, <span class="math inline">\(m \geq 9\)</span> and rational functions with degree sum 4, we explicitly find at most 13 fields in which the desired pair may not exist.</p>