Three Dimensional Strongly Symmetric Circulant Tensors

TL;DR

This paper establishes a precise criterion for when a specific class of three-dimensional, strongly symmetric circulant tensors are positive semi-definite and explores their sum-of-squares property through theoretical and numerical analysis.

Contribution

It provides a necessary and sufficient condition for positive semi-definiteness of these tensors and investigates their sum-of-squares status.

Findings

Condition is both necessary and sufficient for positive semi-definiteness.

Numerical tests support the sufficiency of the condition for sum-of-squares.

Results advance understanding of tensor positivity and sum-of-squares properties.

Abstract

In this paper, we give a necessary and sufficient condition for an even order three dimensional strongly symmetric circulant tensor to be positive semi-definite. In some cases, we show that this condition is also sufficient for this tensor to be sum-of-squares. Numerical tests indicate that this is also true in the other cases.