  The degree of the generators of invariant polynomial rings of is a long
standing open problem since the very initial study of the invariant theory in
the 19th century. Motivated by its significant role in characterizing
multipartite entanglement, we study the invariant polynomial rings of local
unitary group---the tensor product of unitary group, and local general linear
group---the tensor product of general linear group. For these two groups, we
prove polynomial upper bounds on the degree of the generators of invariant
polynomial rings. On the other hand, systematic methods are provided to to
construct all homogenous polynomials that are invariant under these two groups
for any fixed degree. Thus, our results can be regarded as a complete
characterization of the invariant polynomial rings. As an interesting
application, we show that multipartite entanglement is additive in the sense
that two multipartite states are local unitary equivalent if and only if
$r$-copies of them are LU equivalent for some $r$.
