  We study the ring R(n,m) of invariants for the left-right action of SL_n
\times SL_n on m-tuples of n by n complex matrices. We show that R(3,m) is
generated by invariants of degree less equal 309 for all m. Then, we use a
combinatorial description of the invariants to show that R(n,m) cannot be
generated by invariants of degree < n^2 for large m. We also compute the
Hilbert series for several cases.
