  We introduce a new criterion which tests if a given decomposition of a given
ternary form $T$ of even degree is unique. The criterion is based on the
analysis of the Hilbert function of the projective set of points $Z$ associated
to the decomposition, and on the Terracini's Lemma which describes tangent
spaces to secant varieties. The criterion works in a range for the length of
the decomposition which is equivalent to the range in which the reshaped
Kruskal's criterion (see [1]) works. Our criterion determines an algorithm for
the identifiability of $T$ which is sensibly faster than algorithms based on
the reshaped Kruskal's criterion, especially when the set of points $Z$ is not
in general position.
