Minimality and uniqueness for decompositions of specific ternary forms

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