Positive definiteness of a class of cyclic symmetric tensors

TL;DR

This paper establishes necessary and sufficient conditions for the positive definiteness of certain cyclic symmetric tensors, extending to non-cyclic cases, and applies these results to inequalities involving ternary quartic polynomials.

Contribution

It provides new necessary and sufficient conditions for positive definiteness of 4th order symmetric tensors, including cyclic and non-cyclic cases, and applies these to polynomial inequalities.

Findings

Conditions for positive semi-definiteness of 4th order cyclic symmetric tensors.

Conditions for positive definiteness of n-dimensional symmetric tensors.

New inequalities for ternary quartic homogeneous polynomials.

Abstract

For a 4th order 3-dimensional cyclic symmetric tensor, a sufficient and necessary condition is bulit for its positive semi-definiteness. A sufficient and necessary condition of positive definiteness is showed for a 4th order $n$-dimensional symmetric tensor. With the help of such a condition, the positive definiteness of a class of 4th order 3-dimensional cyclic symmetric tensors is given. Moreover, the positive definiteness of a class of non-cyclic symmetric tensors is showed also. By applying these conclusions, several (strict) inequalities are erected for ternary quartic homogeneous polynomials.