Copositivity of Three-Dimensional Symmetric Tensors

Liqun Qi (), Yisheng Song () and Xinzhen Zhang
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Liqun Qi: School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, P. R. China
Yisheng Song: School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P. R. China
Xinzhen Zhang: School of Mathematics, Tianjin University, Tianjin 300354, P. R. China

Asia-Pacific Journal of Operational Research (APJOR), 2023, vol. 40, issue 03, 1-16

Abstract: In this paper, we seek analytically checkable necessary and sufficient condition for copositivity of a three-dimensional symmetric tensor. We first show that for a general third-order three-dimensional symmetric tensor, checking copositivity is equivalent to solving a quartic equation and some quadratic equations. All of them can be solved analytically. Thus, we present an analytical way to check copositivity of a third-order three-dimensional symmetric tensor. Then, we consider a model of vacuum stability for ℤ3 scalar dark matter. This is a special fourth-order three-dimensional symmetric tensor. We show that an analytically expressed necessary and sufficient condition for this model bounded from below can be given, by using a result given by Ulrich and Watson in 1994.

Keywords: Copositive tensors; symmetric tensors; analytically checkable; vacuum stability (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0217595922500324

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