Optimal $ Z $-eigenvalue inclusion intervals of tensors and their applications

<p style='text-indent:20px;'>Firstly, a weakness of Theorem 3.2 in [Journal of Industrial and Management Optimization, 17(2) (2021) 687-693] is pointed out. Secondly, a new Geršgorin-type <inline-formula><tex-math id="M2">\begin{document}$ Z $\end{document}</tex-math></inline-formula>-eigenvalue inclusion interval for tensors is given. Subsequently, another Geršgorin-type <inline-formula><tex-math id="M3">\begin{document}$ Z $\end{document}</tex-math></inline-formula>-eigenvalue inclusion interval with parameters for even order tensors is presented. Thirdly, by selecting appropriate parameters some optimal intervals are provided and proved to be tighter than some existing results. Finally, as an application, some sufficient conditions for the positive definiteness of homogeneous polynomial forms as well as the asymptotically stability of time-invariant polynomial systems are obtained. As another application, bounds of <inline-formula><tex-math id="M4">\begin{document}$ Z $\end{document}</tex-math></inline-formula>-spectral radius of weakly symmetric nonnegative tensors are presented, which are used to estimate the convergence rate of the greedy rank-one update algorithm and derive bounds of the geometric measure of entanglement of symmetric pure state with nonnegative amplitudes.</p>