Catalecticant intersections and confinement of decompositions of forms

TL;DR

This paper introduces the concept of confinement of decompositions for forms, providing new proofs and computational results on the number of decompositions for plane cubics and quartics.

Contribution

It presents the novel notion of confinement for decompositions, simplifying the analysis of possible decompositions and applying it to classical and new cases.

Findings

Shorter proof that 3 general plane cubics have 2 decompositions of rank 6

Computed that 4 general plane quartics have 18 decompositions of rank 10

Introduced the technique of confinement to reduce parameters in decomposition problems

Abstract

We introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds, lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data. With the technique of confinement, we obtain here two results. First, we give a new, shorter proof of a result by London (\cite{London90}) that $3$ general plane cubics have $2$ simultaneous Waring decompositions of rank $6$. Then we compute, with the software Bertini, that $4$ general plane quartics have $18$ different decompositions of rank $10$ (a result which was not known before).