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

TL;DR

This paper extends lattice-theoretic methods to establish degree bounds for rational generators of invariant fields under finite abelian group actions, building on prior work on polynomial generators.

Contribution

It adapts existing lattice-based techniques to derive degree bounds for rational invariant generators, expanding the scope from polynomial to rational invariants.

Findings

Degree bounds for rational invariants are similar to those for polynomial invariants.

Lattice-theoretic methods are effective for analyzing rational invariant generators.

The approach requires only minor modifications from previous polynomial cases.

Abstract

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.