The analytic criterion of strict copositivity for a 4th-order 3-dimensional tensor

TL;DR

This paper establishes a necessary and sufficient criterion for the strict copositivity of 4th-order 3-dimensional symmetric tensors, enabling effective verification of such tensors with specific entry constraints.

Contribution

It provides the first complete characterization of strict copositivity for 4th-order 3D symmetric tensors, including those with entries ±1 or 0.

Findings

Derived necessary and sufficient conditions for strict copositivity.

Extended criteria to tensors with entries ±1 and 0.

Facilitated verification of general 4th-order 3D symmetric tensors.

Abstract

This paper focuses on the strict copositivity analysis of 4th-order 3-dimensional symmetric tensors. A necessary and sufficient condition is provided for the strict copositivity of a fourth-order symmetric tensor. Subsequently, building upon this conclusion, we discuss the strict copositivity of fourth-order three-dimensional symmetric tensors with its entries $\pm 1, 0$, and further build their necessary and sufficient conditions. Utilizing these theorems, we can effectively verify the strict copositivity of a general fourth-order three-dimensional symmetric tensors.