A remark on Waring decompositions of some special plane quartics

TL;DR

This paper investigates specific Waring decompositions of plane quartics with high rank, revealing geometric conditions under which certain decompositions imply tangency to a conic.

Contribution

It establishes a geometric criterion linking Waring decompositions of quartics to tangency conditions on conics, expanding understanding of quartic decompositions.

Findings

Decomposition involving seven linear forms and a quadratic form implies tangency to a conic.

The intersection points of lines with a cubic form determine tangency conditions.

Provides conditions under which a line is tangent to a conic in the context of quartic decompositions.

Abstract

This work concerns Waring decompositions of a certain kind of plane quartics of high rank. The main result is the following. Let x, l_1, ...., l_7 be linear forms and q a quadratic form on a vector space of dimension 3. If x^2q=l_1^4+...+l_7^4 and the lines l_1=0, ..., l_7=0 in P^2 intersect x=0 in seven distinct points, then the line x=0 is (possibly improperly) tangent to the conic q=0.