On generating the ring of matrix semi-invariants

Understand

For a field $\mathbb{F}$, let $R(n, m)$ be the ring of invariant polynomials for the action of $\mathrm{SL}(n, \mathbb{F}) \times \mathrm{SL}(n, \mathbb{F})$ on tuples of matrices -- $(A, C)\in\mathrm{SL}(n, \mathbb{F}) \times \mathrm{SL}(n, \mathbb{F})$ sends $(B_1, \dots, B_m)\in M(n, \mathbb{F})^{\oplus m}$ to $(AB_1C^{-1}, \dots, AB_mC^{-1})$.

• In this paper we call $R(n, m)$ the \emph{ring of matrix semi-invariants}.
• Let $\beta(R(n, m))$ be the smallest $D$ s.t.
• matrix semi-invariants of degree $\leq D$ generate $R(n, m)$.