On generating the ring of matrix semi-invariants

For a field F, let R(n,m) be the ring of invariant polynomials for the action of SL(n,F) × SL(n,F) on tuples of matrices – (A,C) ∈ SL(n,F)× SL(n,F) sends (B1, . . . , Bm) ∈ M(n,F) to (AB1C , . . . , ABmC ). In this paper we call R(n,m) the ring of matrix semi-invariants. Let β(R(n,m)) be the smallest D s.t. matrix semi-invariants of degree ≤ D generate R(n,m). Guided by the Procesi-Razmyslov-Formanek approach of proving a strong degree bound for generating matrix invariants, we exhibit several interesting structural results for the ring of matrix semi-invariants R(n,m) over fields of characteristic 0. Using these results, we prove that β(R(n,m)) = Ω(n), and β(R(2,m)) ≤ 4.