  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 $8$.
