## 5 Numerical Radius of Real and Imaginary Parts of \(C(p)\)  
The numerical radius of real and imaginary parts of \(C(p)\) is now investigated. To this end, let \(T\in\mathcal{M}_{n}(\mathbb{C})\) with the Cartesian decomposition \(T=P+iQ\). Then,  
\[W(T)\subseteq W(P)+W(Q).\]  
Hence,  
\[\sigma(T)\subseteq[\lambda_{\min}(P),\lambda_{\max}(P)]\times[\lambda_{\min}( Q),\lambda_{\max}(Q)].\]  
In case of the companion matrix \(C(p)\), it follows that all the zeros of \(p(z)\) in (4) are located in the rectangle  
\[[\lambda_{\min}(\text{Re}[C(p)]),\lambda_{\max}(\text{Re}[C(p)])]\times[ \lambda_{\min}(\text{Im}[C(p)]),\lambda_{\max}(\text{Im}[C(p)])]. \tag{19}\]  
In [14], [PERSON] provided an explicit formula for the characteristic polynomial of \(\text{Re}[C(p)]\), which is given as \begin{table}
\begin{tabular}{c c} \hline \hline
**Mathematician** & **Upper Bound** \\ \hline [PERSON] (6) & 5 \\ [PERSON] and [PERSON] (7) & 5.860057831 \\ [PERSON] (8) & 12.58333333 \\ [PERSON] and [PERSON] (9) & 18.19610776 \\ [PERSON] (10) & 17.44802607 \\ [PERSON] (11) & 5.845408848 \\ [PERSON] (12) & 4.040959271 \\ [PERSON] and [PERSON] (13) & 4.916052295 \\ [PERSON] et al. (14) & 4.867955746 \\
**Corollary 1** & **3.941508802** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparisons of upper bounds based on the considered polynomial \(q_{1}(z)\). \[p_{\text{Real}}(z):=(z+\text{Re}(a_{n}))\prod_{j=1}^{n-1}\left(z-\cos \left(\frac{j\pi}{n}\right)\right)-\sum_{j=1}^{n-1}\left(\prod_{k\
eq j}^{n-1} \left(z-\cos\left(\frac{k\pi}{n}\right)\right)\right)|v_{j}|^{2},\]  
where  
\[v_{j}=\frac{1}{\sqrt{2n}}\Bigg{[}(1-\overline{a}_{n-1})\sin \left(\frac{j\pi}{n}\right)-\sum_{k=2}^{n-1}\overline{a}_{n-k}\sin\left(\frac {kj\pi}{n}\right)\Bigg{]}.\]  
In the same work [14], an explicit rectangle that contains the rectangle (19), and thus that contains all the zeros of \(p(z)\), is obtained in the result below. **Theorem 2** ([14], [PERSON]). : _If \(z\) is any zero of \(p(z)=z^{n}+c_{n}z^{n-1}+\cdots+c_{2}z+c_{1}\), \((c_{1}\
eq 0)\), then \(z\) belongs to the rectangle \([-c,c]\times[-d,d]\), where_  
\[c=\frac{1}{2}\Bigg{[}|\text{Re}(c_{n})|+\cos\Big{(}\frac{\pi}{n} \Big{)}+\sqrt{\left(|\text{Re}(c_{n})|-\cos\Big{(}\frac{\pi}{n}\Big{)}\right)^ {2}+|c_{n-1}-1|^{2}+\sum_{k=1}^{n-2}|c_{k}|^{2}}\Bigg{]}\]  
_and_  
\[d=\frac{1}{2}\Bigg{[}|\text{Im}(c_{n})|+\cos\Big{(}\frac{\pi}{n }\Big{)}+\sqrt{\left(|\text{Im}(c_{n})|-\cos\Big{(}\frac{\pi}{n}\Big{)} \right)^{2}+|c_{n-1}-1|^{2}+\sum_{k=1}^{n-2}|c_{k}|^{2}}\Bigg{]}.\]  
Based on the results obtained in this work, in what follows, we provide another possible rectangle. In order to establish our result, we need the following lemma (see [21]):  
**Lemma 5**. : _Let \(A,B,C,D\in\mathscr{B}(\mathscr{H})\) and \(T=\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]\). Then,_  
\[\omega(T)\leq\frac{1}{2}\bigg{(}\omega(A)+\omega(D)+\sqrt{\left( \omega(A)-\omega(D)\right)^{2}+\left(\omega(B+C)+\omega(B-C)\right)^{2}} \bigg{)}.\]  
Now we can give our explicit rectangle, which contains the rectangle (19) and thus all of the zeros of \(p(z)\). **Theorem 3**. : _Let \(q(z)\) be any even complex polynomial as given in (15) whose degree is \(\geq 4\). If \(z\) is any zero of \(q(z)\), then \(z\) belongs to the rectangle \([-s,s]\times[-t,t]\), where_  
_and_  
\[J :=\sqrt{\Big{(}|\text{Im}(a_{2n})|-\cos\Big{(}\frac{\pi}{n}\Big{)} \Big{)}^{2}+|1+a_{2n-1}|^{2}+\sum_{k=n+1}^{2n-2}|a_{k}|^{2}}.\]  
Proof. : Employing Lemma 5, on the real part of \(C(q)\) (17), which is obtained in the previous section, by setting \(A=P_{11}\), \(B=P_{12}\), \(C=P_{21}\) and \(D=P_{22}\), it is enough to show that  
\[\omega(\text{Re}[C(q)])\] \[\leq\frac{1}{2}\Big{(}\omega(P_{11})+\omega(P_{22})+\sqrt{\big{(} \omega(P_{11})-\omega(P_{22})\big{)}^{2}+\big{(}\omega(P_{12}+P_{21})+\omega( P_{12}-P_{21})\big{)}^{2}}\Big{)}. \tag{20}\]  
Let us simplify that. Indeed, we have  
\[P_{11}=\left[