On the separating Noether number of finite abelian groups

TL;DR

The paper addresses the problem of determining the separating Noether number $β_{sep}(G)$ for finite abelian groups and connects it to zero-sum theory through the Davenport constant. Using additive combinatorics methods, it proves an exact formula under a key hypothesis on $D(n_sG)$ and derives explicit corollaries for $p$-groups and for ranks $2$, $3$, and $5$. The main result shows $β_{sep}(G)=n_s+n_{s+1}+\dots+n_r$ when the rank is odd, and provides tight upper bounds when the rank is even, with concrete expressions in the corollaries. This work strengthens the bridge between invariant theory and zero-sum theory, offering precise separating-invariant thresholds for broad families of finite abelian groups and informing the structure of separating sets of invariants.

Abstract

The separating Noether number $β_{\mathrm{sep}}(G)$ of a finite group $G$ is the minimal positive integer $d$ such that for every finite $G$-module $V$ there is a separating set consisting of invariant polynomials of degree at most $d$. In this paper we use methods from additive combinatorics to investigate the separating Noether number for finite abelian groups. Among others, we obtain the exact value of $β_{\mathrm{sep}}(G)$, provided that $G$ is either a $p$-group or has rank $2$, $3$ or $5$.

Theorems & Definitions (9)

• proof
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['main']}
• claim A1
• proof : Proof of \ref{['cl1']}
• proof : Proof of Corollary \ref{['co1']}
• proof : Proof of Corollary \ref{['co2']}