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<p>In this paper, we mainly discuss the non-negativity conditions for quartic homogeneous polynomials with 3 variables, which is the analytic conditions of copositivity of a class of 4th order 3-dimensional symmetric tensors. For a 4th order 3-dimensional symmetric tensor with its entries 1 or <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, an analytic necessary and sufficient condition is given for its strict copositivity with the help of the properties of strictly semi-positive tensors. Moreover, a necessary and sufficient condition is established for copositivity of such a tensor also. Several (strict) inequalities of ternary quartic homogeneous polynomial are established by means of these analytic conditions.</p>

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Copositivity criteria of a class of fourth order symmetric tensors

  • 期刊论文

  • 发表日期：

    2026年2月11日

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  • Yisheng Song,
  • Jinjie Liu

摘要

In this paper, we mainly discuss the non-negativity conditions for quartic homogeneous polynomials with 3 variables, which is the analytic conditions of copositivity of a class of 4th order 3-dimensional symmetric tensors. For a 4th order 3-dimensional symmetric tensor with its entries 1 or \(-1\) - 1 , an analytic necessary and sufficient condition is given for its strict copositivity with the help of the properties of strictly semi-positive tensors. Moreover, a necessary and sufficient condition is established for copositivity of such a tensor also. Several (strict) inequalities of ternary quartic homogeneous polynomial are established by means of these analytic conditions.