# On the identifiability of ternary forms

## TL;DR

This paper introduces a new Hilbert function-based method to determine the minimality and identifiability of ternary forms' Waring decompositions, applicable across all decomposition lengths, including the generic rank.

## Contribution

The paper presents a novel, comprehensive approach for assessing identifiability of ternary forms that works for all decomposition lengths, including the generic rank, especially demonstrated for degree 8 forms.

## Key findings

- Method distinguishes between identifiable and non-identifiable points.
- Applicable for all decomposition lengths from 2 up to the generic rank.
- Successfully described in detail for degree 8 forms.

## Abstract

We describe a new method to determine the minimality and identifiability of a Waring decomposition $A$ of a specific form (symmetric tensor) $T$ in three variables. Our method, which is based on the Hilbert function of $A$, can distinguish between forms in the span of the Veronese image of $A$, which in general contains both identifiable and not identifiable points, depending on the choice of coefficients in the decomposition. This makes our method applicable for all values of the length $r$ of the decomposition, from $2$ up to the generic rank, a range which was not achievable before. Though the method in principle can handle all cases of specific ternary forms, we introduce and describe it in details for forms of degree $8$.