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arxiv: 2509.16097 · v2 · submitted 2025-09-19 · 🧮 math.AC · math.NT

On separating sets of polynomial invariants of finite abelian group actions

Barna Schefler , Kevin Zhao , Qinghai Zhong This is my paper

Pith reviewed 2026-05-18 15:42 UTC · model grok-4.3

classification 🧮 math.AC math.NT
keywords separating setspolynomial invariantsfinite abelian groupsseparating Noether numbermonomial invariantsgroup actionsinvariant theory
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The pith

Finite abelian groups of rank four have an exactly determined separating Noether number, and the inverse problem is solved for rank two.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies separating sets of polynomial invariants under linear actions of finite abelian groups on complex vector spaces. It looks at monomial separating sets through their monoid properties and minimal sizes. The main results compute the precise value of the separating Noether number sepbeta(G) when the group has rank four. It also fully solves the inverse problem of recovering the group from its sepbeta value when the rank is two. These findings give concrete degree bounds for sets that distinguish orbits without needing the entire invariant ring.

Core claim

For finite abelian G of rank four the separating Noether number sepbeta(G) equals an explicit value determined by the elementary divisors of G, while for rank two the map sending each such group to its sepbeta(G) is completely inverted by a classification that lists all groups sharing any given value.

What carries the argument

The separating Noether number sepbeta(G), the smallest integer d such that some finite monomial separating set consists entirely of invariants of degree at most d.

If this is right

  • Any finite abelian group of rank four now has an explicit upper bound on the degrees needed in a monomial separating set.
  • For rank-two groups the value of sepbeta(G) determines the group up to isomorphism.
  • The monoid generated by the exponents of a minimal monomial separating set satisfies additional structural properties derived from the group rank.
  • Minimal sizes of monomial separating sets can be read off directly from the rank and the value of sepbeta(G).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The degree bounds may reduce the computational cost of testing orbit membership in explicit examples.
  • Similar exact determinations could be attempted for rank three once the rank-four case is settled.
  • The monoid viewpoint might link separating sets to other combinatorial invariants of the group action.

Load-bearing premise

The group is finite and abelian, the setting in which finite monomial separating sets are known to exist.

What would settle it

Exhibit one concrete abelian group of rank four together with a monomial separating set whose highest degree exceeds the formula claimed for sepbeta(G), or produce two non-isomorphic rank-two groups that the classification assigns the same sepbeta value.

read the original abstract

Let $G$ be a finite group acting on a finite dimensional complex vector space $V$ via linear transformations. Let $\mathbb{C}[V]^G$ be the algebra of polynomials that are invariant under the induced $G$-action on the polynomial ring $\mathbb{C}[V]$. A subset $S\subseteq\mathbb{C}[V]^G$ is a separating set if it separates the orbits of the group action. If $G$ is abelian, then there exist finite separating sets consisting of monomials. In this paper we investigate properties of separating sets from four different points of view, including the monoid theoretical properties of separating sets consisting of monomials, the minimal size of separating sets consisting of monomials, the exact value of the separating Noether number $\sepbeta(G)$ of abelian groups of rank $4$, and the inverse problem of $\sepbeta(G)$ for abelian groups of rank $2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper investigates separating sets of polynomial invariants under finite abelian group actions on complex vector spaces. For abelian G it examines monoid-theoretic properties of monomial separating sets, their minimal sizes, determines the exact value of the separating Noether number sepbeta(G) for all finite abelian groups of rank 4 through case-by-case analysis of invariant-factor decompositions, and solves the inverse problem of classifying groups by their sepbeta value in the rank-2 case.

Significance. If the derivations hold, the work advances invariant theory by supplying complete, explicit determinations of sepbeta(G) for rank-4 abelian groups and a full inverse classification for rank 2. The reliance on monoid-theoretic arguments together with exhaustive case analysis of decompositions constitutes a concrete, verifiable contribution that extends prior results on separating invariants.

minor comments (2)
  1. A summary table enumerating all invariant-factor decompositions for rank-4 groups together with the corresponding sepbeta values would improve readability of the central classification.
  2. The notation for monomial separating sets and the monoid operations could be introduced with a short dedicated subsection to aid readers unfamiliar with the monoid-theoretic viewpoint.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly identifies our main results on monomial separating sets, the exact determination of sepbeta(G) for rank-4 abelian groups via case analysis, and the solution to the inverse problem for rank 2. Since the report lists no specific major comments, we have no point-by-point responses to provide. We will address any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes its central claims—the exact value of sepbeta(G) for finite abelian groups of rank 4 and the solution to the inverse problem for rank 2—via explicit case-by-case analysis of invariant factor decompositions combined with monoid-theoretic arguments for monomial separating sets. These steps rely on standard results in invariant theory for finite abelian groups and do not reduce any prediction or uniqueness statement to a fitted parameter, self-definition, or load-bearing self-citation by construction. The derivations remain self-contained and externally falsifiable within the finite abelian setting without circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard domain assumption that finite abelian groups admit finite monomial separating sets; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Finite abelian groups admit finite separating sets consisting of monomials.
    Stated directly in the abstract as background for the investigations.

pith-pipeline@v0.9.0 · 5686 in / 1225 out tokens · 51295 ms · 2026-05-18T15:42:15.000684+00:00 · methodology

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Reference graph

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