When is a ring of torus invariants a polynomial ring? (Q1328166)

Let \(T\) be an algebraic torus over the algebraically closed field \(k\) and let \(\rho:T \to GL(V)\) be a finite dimensional rational representation. The author develops criteria to determine when the ring of invariants \(R=k[V]^ T\) is a polynomial ring. He does this by first determining criteria for \(R\) to have a ``monomial system of parameters'' (invariant monomials algebraically independent over \(k\) such that \(R\) is integral over the algebra they generate) and then criteria for the integral extension to be trivial. The first condition is constructive in terms of the weight lattice associated to the representation and various equivalent formulations. There is a variety of formulations given for the second, including a completely algorithmic one.