Strict Copositivity for a Class of 3rd Order Symmetric Tensors

TL;DR

This paper establishes necessary and sufficient conditions for strict copositivity of a specific class of third order three-dimensional symmetric tensors with entries in {-1,0,1}, using polynomial decomposition methods.

Contribution

It provides new analytic criteria for strict copositivity of 3rd order 3D symmetric tensors, including special cases with entries in {-1,0,1} and general tensors.

Findings

Derived necessary and sufficient conditions for copositivity.

Presented inequalities for cubic ternary homogeneous polynomials.

Established criteria for general 3rd order 3D tensors.

Abstract

In this article, we mainly give the strictly copositive conditions of a special class of third order three dimensional symmetric tensors. More specifically, by means of the polynomial decomposition method, the analytic sufficient and necessary conditions are established for checking the strict copositivity of a 3rd order 3-dimensional symmetric tensor with its entries in $\{-1,0,1\}$. Several strict inequalities of cubic ternary homogeneous polynomials are presented by applying these conclusions. Some criteria which ensure the strict copositivity of a general 3rd order 3-dimensional tensor are obtained