Note on Z -eigenvalue inclusion theorems for tensors

Abstract

<p style='text-indent:20px;'>Wang et al. gave four <inline-formula><tex-math id="M2">\begin{document}$ Z $\end{document}</tex-math></inline-formula>-eigenvalue inclusion intervals for tensors in [Discrete and Continuous Dynamical Systems Series B, 1 (2017), 187-198]. However, these intervals always include zero, and hence could not be used to identify the positive definiteness of a homogeneous polynomial form. In this note, we present a new <inline-formula><tex-math id="M3">\begin{document}$ Z $\end{document}</tex-math></inline-formula>-eigenvalue inclusion interval with parameters for even-order tensors, which not only overcomes the above shortcomings under certain conditions, but also provides a checkable sufficient condition for the positive definiteness of homogeneous polynomial forms, as well as the asymptotically stability of time-invariant polynomial systems.