A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form
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 penc...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/2339746 |
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| author | Yang Guo |
| author_facet | Yang Guo |
| author_sort | Yang Guo |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-f0597163d1de496fbd695fd9ac6b7222 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-f0597163d1de496fbd695fd9ac6b72222025-02-03T01:21:45ZengWileyJournal of Mathematics2314-47852021-01-01202110.1155/2021/2339746A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic FormYang Guo0Department of MathematicsIn 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.http://dx.doi.org/10.1155/2021/2339746 |
| spellingShingle | Yang Guo A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form Journal of Mathematics |
| title | A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form |
| title_full | A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form |
| title_fullStr | A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form |
| title_full_unstemmed | A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form |
| title_short | A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form |
| title_sort | novel necessary and sufficient condition for the positivity of a binary quartic form |
| url | http://dx.doi.org/10.1155/2021/2339746 |
| work_keys_str_mv | AT yangguo anovelnecessaryandsufficientconditionforthepositivityofabinaryquarticform AT yangguo novelnecessaryandsufficientconditionforthepositivityofabinaryquarticform |