Minimality and uniqueness of decompositions of specific ternary forms

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 phenomenon 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.