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

A novel necessary and sufficient condition for the positivity of a binary quartic form