A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form. (30th November 2021)

• Record Type: Journal Article
• Authors: Guo, Yang
• Other Names: Hong Shaofang Academic Editor.
• Is Part Of: Journal of mathematics. Volume 2021(2021)
• Publication Date: 2021-11-30
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1155/2021/2339746
• Languages: English
• ISSNs: 2314-4629

Abstract

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.