A Finiteness Property of Torus Invariants

In this paper the invariant subring $R_n$ of an algebraic torus $T=(\mathbb{C}^\times)^r$ acting on the multi-homogenous polynomial ring $$S^{\boxtimes n}=\bigoplus_{d=0}^\infty (S^{(d)})^{\otimes n},$$ where $S^{(d)}$ is the $d$th graded piece of the polynomial ring $S=\mathbb{C}[x_1,\dots,x_k]$, is studied from the viewpoint of matrices whose entries sum to zero. Using these weight matrices we prove that there exists a $d_1$ such that for all positive integers $n$, the relations of the invariant subring $R_n$ are generated in multi-homogenous degree $\leq d_1$. Grant: 0943832