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

We study the ring R ( n , m ) of invariants for the left–right action of SL n × SL n on m -tuples of n × n complex matrices. We show that 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 ) cannot be generated by invariants of degree < n 2 for large m . We also compute the Hilbert series for several cases. [ABSTRACT FROM AUTHOR]