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

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.

Positive definiteness for 4th order symmetric tensors and applications

“…In 1998, Fu [11] pointed out that Hasan-Hasan's results are sufficient only. Recently, Guo [12] showed a new necessary and sufficient condition without the discriminant. Very recently, Qi-Song-Zhang [13] gave a new necessary and sufficient condition other than the above results.…”