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A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form

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  • Yang Guo
  • Shaofang Hong

Abstract

In this paper, by considering the common points of two conics instead of the roots of the binary quartic form, we propose a novel necessary and sufficient condition for the positivity of a binary quartic form using the theory of the pencil of conics. First, we show the degenerate members of the pencil of conics according to the distinct natures of the common points of two base conics. Then, the inequalities about the parameters of the degenerate members are obtained according to the properties of the degenerate conics. Last, from the inequalities, we derive a novel criterion for determining the positivity of a binary quartic form without the discriminant.

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  • Yang Guo & Shaofang Hong, 2021. "A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form," Journal of Mathematics, Hindawi, vol. 2021, pages 1-7, November.
    • Handle: RePEc:hin:jjmath:2339746
    DOI: 10.1155/2021/2339746
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    Cited by:

    1. Yisheng Song & Xudong Li, 2022. "Copositivity for a Class of Fourth-Order Symmetric Tensors Given by Scalar Dark Matter," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 334-346, October.

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