  The equivalence problem under local unitary transformation for $n$--partite
pure states is reduced to the one for $(n-1)$--partite mixed states. In
particular, a tripartite system
$\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$, where $\mathcal{H}_j$
is a finite dimensional complex Hilbert space for $j=A,B,C$, is considered and
a set of invariants under local transformations is introduced, which is
complete for the set of states whose partial trace with respect to
$\mathcal{H}_A$ belongs to the class of generic mixed states.
