Degree bounds for rational generators of invariant fields of finite abelian groups

We study degree bounds on rational but not necessarily polynomial generators for the field $\mathbf{k}(V{)}^G$ of rational invariants of a linear action of a finite abelian group. We show that lattice-theoretic methods used recently by the author and collaborators to study polynomial generators for the same field largely carry over, after minor modifications to the arguments. It then develops that the specific degree bounds found in that setting also carry over.