Minimality and uniqueness for decompositions of specific ternary forms

TL;DR

This paper investigates the minimality and uniqueness of Waring decompositions for specific ternary forms, introducing an algebraic approach using Hilbert-Burch matrices and liaison to analyze and find alternative decompositions.

Contribution

It develops a new criterion based on Hilbert-Burch matrices for testing minimality and uniqueness of ternary form decompositions, with an explicit algorithm for degree 9 forms.

Findings

The criterion effectively detects minimality and uniqueness in ternary forms.

Liaison techniques can find alternative, shorter decompositions when non-uniqueness occurs.

A new phenomenon is observed in degree 9 forms where points of different ranks coexist in the span.

Abstract

The paper deals with the computation of the rank and the identifiability of a specific ternary form. Often, one knows some short Waring decomposition of a given form, and the problem is to determine whether the decomposition is minimal and unique. We show how the analysis of the Hilbert-Burch matrix of the set of points representing the decomposition can solve this problem in the case of ternary forms. Moreover, when the decomposition is not unique, we show how the procedure of liaison can provide alternative, maybe shorter, decompositions. We give an explicit algorithm that tests our criterion of minimality for the case of ternary forms of degree $9$. This is the first numerical case in which a new phaenomenon appears: the span of $18$ general powers of linear forms contains points of (subgeneric) rank $18$, but it also contains points whose rank is $17$, due to the existence of a second shorter decomposition which is completely different from the given one.