The maximum and minimum value of homogeneous polynomial under different norms via tensors

Abstract

For any homogeneous polynomial, it can be expressed as the product of a tensor $\mathcal{A}$ and a vector $x$, we denote it by $P_{\mathcal{A}}(x)$. With the change of the norm of $x$, the maximum value (resp. the minimum value) of $P_{\mathcal{A}}(x)$ is changed. In this paper, by the properties of tensor $\mathcal{A}$, we study the relationships between the maximum values (resp. minimum values) of $P_{\mathcal{A}}(x)$ under different norms of $x$. We present that the maximum values (resp. the minimum values) of $P_{\mathcal{A}}(x)$ at different norms of $x$ always have the same sign. Moreover, the relationship between the magnitudes of the maximum values (resp. the minimum values) of $P_{\mathcal{A}}(x)$ at different norms of $x$ are characterized.
Further, some inequalities on H-eigenvalues and Z-eigenvalues of tensor $\mathcal{A}$ are obtained directly. And some applications on definite positive of tensors and hypergraphs are given.