Hilbert series and degree bounds for matrix (semi-)invariants

We study the ring R(n,m)R(n,m) of invariants for the left–right action of SLn×SLnSLn×SLn on m  -tuples of n×nn×n complex matrices. We show that R(3,m)R(3,m) is generated by invariants of degree ≤309 for all m  . Then, we use a combinatorial description of the invariants to show that R(n,m)R(n,m) cannot be generated by invariants of degree