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Strict Copositivity for a Class of 3rd Order Symmetric Tensors

Min Li, Yisheng Song

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

Strict Copositivity for a Class of 3rd Order Symmetric 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 . 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

Paper Structure

This paper contains 3 sections, 12 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strict Copositivity of Third Order Three-Dimensional Symmetric Tensors

Key Result

Theorem 2.2

(SH1988QSZ2022) Suppose that $\mathcal{A} = (a_{ijk}) \in S_{3, 2}$. Then $\mathcal{A}$ is (strictly) copositive if and only if $a_{111} \ge 0\ (> 0)$, $a_{222} \ge 0\ (> 0)$ and

Theorems & Definitions (13)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Corollary 3.5
  • Corollary 3.6
  • ...and 3 more