Generic orbits, normal bases, and generation degree for fields of rational invariants

TL;DR

This paper establishes bounds relating the Noether number and spanning degree for rational invariants of finite groups, generalizing recent results and exploring their properties in invariant theory.

Contribution

It introduces new inequalities connecting the Noether number and spanning degree, extending previous work to more general settings and analyzing their behavior.

Findings

Proves $eta_{field} \, extless= 2D_{span} + 1$, which is sharp.

Shows $D_{span}$ is related to known quantities in invariant theory.

Establishes inequalities for $D_{span}$ without the coprime characteristic assumption.

Abstract

For a faithful linear representation $V$ of a finite group $G$ in coprime characteristic, we show that if the field Noether number $\beta_{\mathrm{field}}$ is the minimum $d$ such that the invariant polynomials of degree $\leq d$ generate the field $k(V)^G$ of rational invariants as a field, and the spanning degree $D_\mathrm{span}$ is the minimum $d$ such that the polynomials of degree $\leq d$ span the rational function field $k(V)$ as a vector space over $k(V)^G$, then $\beta_{\mathrm{field}} \leq 2D_\mathrm{span} + 1$, and this is sharp. This generalizes a recent result of Edidin and Katz. We also study $D_\mathrm{span}$. We show that it is related to various quantities previously studied in invariant and representation theory. Dropping the coprime characteristic hypothesis, we prove several basic inequalities, including that it is monotonically nondecreasing in $G$, nonincreasing in $V$, and satisfies $D_\mathrm{span} \leq |G|-1$. The latter refines a recent result of Kollar and Pham.