On the study of decompositions of forms in four variables

TL;DR

This paper investigates the structure of sextic forms in four variables with a specific decomposition length, identifying a geometric subvariety containing all non-identifiable forms and providing an algorithm to determine identifiability.

Contribution

It introduces a geometric description of the subvariety of non-identifiable sextic forms in four variables and proposes an algorithm to verify form identifiability.

Findings

Identifies a closed subvariety containing all non-identifiable sextic forms in four variables.

Provides a geometric characterization of this subvariety.

Develops an algorithm to determine if a given form is identifiable.

Abstract

In the space of sextic forms in 4 variables with a decomposition of length 18 we determine and describe a closed subvariety which contains all non-identifiable sextics. The description of the subvariety is geometric, but one can derive from that an algorithm which can guarantee that a given form is identifiable.