二元四次型为正的一个新的充要条件
A novel necessary and sufficient condition for the positivity of a binary quartic form
摘要：
本文利用二次曲线束理论，通过考虑两条二次曲线的公共点而不是二元四次型的根，给出了二元四次型为正的一个新的充要条件.首先，根据两条基二次曲线公共点的不同性质，给出了二次曲线束的退化成员。然后，根据退化二次曲线的性质，得到了关于退化杆件参数的不等式。最后，从不等式中我们得到了一个新的判定二元四次型的正性的准则。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
