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

TL;DR

This paper introduces a new necessary and sufficient condition for determining the positivity of binary quartic forms by analyzing the common points of two conics and their degenerate members, avoiding the use of the discriminant.

Contribution

It presents a novel criterion based on the properties of conic pencils, providing an alternative to discriminant-based methods for positivity testing.

Findings

Derived inequalities for degenerate conics in the pencil

Established a criterion for positivity without using the discriminant

Analyzed common points of two conics to characterize positivity

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.