On the Noether Bound for Noncommutative Rings

Ellen Kirkman; Kewen Peng; Luigi Ferraro; W. Frank Moore

arxiv: 1907.06761 · v1 · pith:KXONGCFJnew · submitted 2019-07-15 · 🧮 math.RA

On the Noether Bound for Noncommutative Rings

Luigi Ferraro , Ellen Kirkman , W. Frank Moore , Kewen Peng This is my paper

Pith reviewed 2026-05-24 20:47 UTC · model grok-4.3

classification 🧮 math.RA

keywords noncommutative ringsinvariant theoryNoether boundcyclic group actionsmatrix representationscharacteristic zerogenerators of invariants

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The pith

Two noncommutative algebras require generators of degree 3n for their invariants under cyclic group actions of order 2n represented by n by n matrices.

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

The paper constructs two specific noncommutative algebras over a field of characteristic zero. Each algebra admits a family of actions by the cyclic group of order 2n, where these actions are implemented via conjugation by n by n matrices. The authors show that in each case the ring of invariants cannot be generated by elements of degree less than 3n. A reader would care because these examples demonstrate that the minimal degree needed for generators of invariants can reach 3n in the noncommutative setting.

Core claim

The paper presents two noncommutative algebras over a field of characteristic zero that each possess a family of actions by cyclic groups of order 2n, represented in n by n matrices, and for which the invariants require generators of degree 3n.

What carries the argument

The two noncommutative algebras together with their invariant subrings under the specified matrix representations of the cyclic group actions.

If this is right

The Noether number for these noncommutative algebras under the given actions is at least 3n.
The bound holds uniformly for the family of actions parameterized by n.
The result applies over any field of characteristic zero.
The two algebras provide distinct examples realizing the same degree requirement.

Where Pith is reading between the lines

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

Similar matrix-representation constructions could be tested on other noncommutative algebras to see whether degree 3n appears for different group orders.
The examples raise the question of whether an upper bound linear in the group order continues to hold in the noncommutative case.
One could check whether the same algebras admit actions by other finite groups that force even higher minimal degrees.

Load-bearing premise

The minimal degree of a generating set for the invariants is exactly 3n rather than some smaller number.

What would settle it

Exhibiting an explicit generating set for the invariants whose elements all have degree strictly less than 3n would falsify the claim for either algebra.

read the original abstract

We present two noncommutative algebras over a field of characteristic zero that each posses a family of actions by cyclic groups of order $2n$, represented in $n \times n$ matrices, requiring generators of degree $3n$.

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.

Desk Editor's Note private letter to a colleague

Paper gives explicit noncommutative algebras where cyclic actions of order 2n force invariant generators of degree exactly 3n, verified by direct linear algebra.

read the letter

The main takeaway is that this paper constructs two explicit noncommutative algebras over a field of characteristic zero, each with a family of cyclic group actions of order 2n realized by n by n matrix representations, and shows that the invariants require generators of degree 3n. That is the concrete new content. The authors supply bases and multiplication rules for the algebras, define the automorphisms explicitly, and check the lower bound by computing homogeneous components degree by degree. Since each check reduces to linear dependence in finite-dimensional vector spaces, the argument is in principle reproducible by hand or machine. They also confirm noncommutativity and faithfulness of the representations by the same direct methods. This explicitness is the paper's strength; it turns an existence claim into something that can be inspected without additional assumptions. The soft spots are limited. The algebras appear tailored to produce the 3n bound, so the work demonstrates that the bound can reach this height but does not yet indicate how often it occurs or whether simpler or more natural examples exist. The focus stays on these two cases rather than a general theory. Readers working on noncommutative invariant theory or Noether bounds will get the most from the concrete constructions and the computational verification. The evidence is solid enough on its own terms that the paper deserves a serious referee, even if revisions might be needed to clarify generality or add context.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs two explicit noncommutative algebras over a field of characteristic zero. For each, it equips the algebra with a family of automorphisms realizing the action of the cyclic group of order 2n via faithful representations by n×n matrices. Direct (though tedious) computation of homogeneous components then verifies that no set of invariants of degree less than 3n generates the full invariant ring.

Significance. If the constructions and verifications hold, the result supplies concrete, checkable examples showing that the Noether bound on the degree of generators for invariant rings can reach 3n in the noncommutative setting. The explicit bases, multiplication rules, and finite-dimensional linear-algebra checks constitute a verifiable existential statement that advances understanding of invariant theory beyond the commutative case.

minor comments (2)

[Abstract] Abstract: 'posses' is a typographical error and should read 'possess'.
[Abstract] The abstract phrasing 'requiring generators of degree 3n' is slightly imprecise; a clearer formulation would state that the minimal degree of any generating set for the invariant ring is exactly 3n.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript. The recommendation for acceptance is appreciated, and we are glad that the explicit constructions are recognized as providing verifiable examples that extend the Noether bound discussion to the noncommutative setting.

Circularity Check

0 steps flagged

No significant circularity; explicit constructions verified by direct computation

full rationale

The paper's central claim is an existence result: two explicitly constructed noncommutative algebras over a field of characteristic zero, equipped with explicit families of automorphisms realizing cyclic group actions of order 2n via n×n matrix representations, are shown by direct (finite-dimensional) computation of homogeneous components to require generators of degree exactly 3n for their invariant rings. No equations reduce a prediction to a fitted input by construction, no self-citation chain bears the load of the main result, and no ansatz or uniqueness theorem is smuggled in. The argument is self-contained against external benchmarks because the algebras, actions, and degree lower bound are all exhibited inside the paper via explicit bases and multiplication rules.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities used in the constructions.

pith-pipeline@v0.9.0 · 5554 in / 1021 out tokens · 21835 ms · 2026-05-24T20:47:30.792109+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Artin and W

M. Artin and W. F. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), no. 2, 171–216

work page 1987
[2]

Benkart and T

G. Benkart and T. Roby, Down-up algebras , J. Algebra 209 (1998), no. 1, 305–344

work page 1998
[3]

Derksen and G

H. Derksen and G. Kemper, Computational Invariant Theory , Encyclopedia of Mathematical Sciences 130: Invariant Theory and Algebraic Transformati on Groups I, Springer-Verlag, Berlin, 2002

work page 2002
[4]

Derksen and J

H. Derksen and J. Sidman, Castelnuovo-Mumford regularity by approximation , Adv. Math. 188 (2004), no. 1, 104–123

work page 2004
[5]

Domokos and P

M. Domokos and P. Heged¨ us, Noether’s bound for polynomial invariants of ﬁnite groups , Arch. Math. 74 (2000), 161–167

work page 2000
[6]

Fleischmann, The Noether bound in invariant theory of ﬁnite groups , Adv

P. Fleischmann, The Noether bound in invariant theory of ﬁnite groups , Adv. Math. 156 (2000), no. 1, 23–32

work page 2000
[7]

Fogarty, On Noether’s bound for polynomial invariants of a ﬁnite grou p, Electron

J. Fogarty, On Noether’s bound for polynomial invariants of a ﬁnite grou p, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 5–7

work page 2001
[8]

Gandini, Ideals of subspace arrangements , Ph.D

F. Gandini, Ideals of subspace arrangements , Ph.D. Thesis, University of Michigan, Ann Arbor, May 2019, ORCID iD: 0000-0002-2619-3555

work page 2019
[9]

Grayson and Michael E

Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/

work page
[10]

Kirkman and J

E. Kirkman and J. Kuzmanovich, Fixed subrings of Noetherian graded regular rings , J. Al- gebra 288 (2005), no. 2, 463–484

work page 2005
[11]

Kirkman, J

E. Kirkman, J. Kuzmanovich, and J. Zhang, Invariants of (-1)-skew polynomial rings under permutation representations, Recent advances in representation theory, quantum groups , al- gebraic geometry, and related topics, 155–192, Contemp. Ma th., vol. 623, Amer. Math. Soc., Providence, RI, 2014

work page 2014
[12]

Kirkman, I

E. Kirkman, I. M. Musson, and D. S. Passman, Noetherian down-up algebras , Proc. Amer. Math. Soc. 127 (1999), no. 11, 3161–3167

work page 1999
[13]

Neusel, Degree bounds: An invitation to postmodern invariant theor y, Topology and its Applications 154 (2007), no

M. Neusel, Degree bounds: An invitation to postmodern invariant theor y, Topology and its Applications 154 (2007), no. 4, 792–814

work page 2007
[14]

Noether, Der endlichkeitssatz der invarianten endlicher gruppen , Math

E. Noether, Der endlichkeitssatz der invarianten endlicher gruppen , Math. Ann 77 (1916), 89–92

work page 1916
[15]

Sezer, Sharpening the generalized Noether bound in the invariant t heory of ﬁnite groups , J

M. Sezer, Sharpening the generalized Noether bound in the invariant t heory of ﬁnite groups , J. Algebra 254 (2002), no. 2, 252–263

work page 2002
[16]

Symonds, On the Castelnuovo-Mumford regularity of rings of polynomi al invariants , An- nals of Math

P. Symonds, On the Castelnuovo-Mumford regularity of rings of polynomi al invariants , An- nals of Math. 174 (2011), 499–517. W ake Forest University, Department of Mathematics and Stati stics, P. O. Box 7388, Winston-Salem, North Carolina 27109 E-mail address : ferrarl@wfu.edu W ake Forest University, Department of Mathematics and Stati stics, P. O. Box ...

work page 2011

[1] [1]

Artin and W

M. Artin and W. F. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), no. 2, 171–216

work page 1987

[2] [2]

Benkart and T

G. Benkart and T. Roby, Down-up algebras , J. Algebra 209 (1998), no. 1, 305–344

work page 1998

[3] [3]

Derksen and G

H. Derksen and G. Kemper, Computational Invariant Theory , Encyclopedia of Mathematical Sciences 130: Invariant Theory and Algebraic Transformati on Groups I, Springer-Verlag, Berlin, 2002

work page 2002

[4] [4]

Derksen and J

H. Derksen and J. Sidman, Castelnuovo-Mumford regularity by approximation , Adv. Math. 188 (2004), no. 1, 104–123

work page 2004

[5] [5]

Domokos and P

M. Domokos and P. Heged¨ us, Noether’s bound for polynomial invariants of ﬁnite groups , Arch. Math. 74 (2000), 161–167

work page 2000

[6] [6]

Fleischmann, The Noether bound in invariant theory of ﬁnite groups , Adv

P. Fleischmann, The Noether bound in invariant theory of ﬁnite groups , Adv. Math. 156 (2000), no. 1, 23–32

work page 2000

[7] [7]

Fogarty, On Noether’s bound for polynomial invariants of a ﬁnite grou p, Electron

J. Fogarty, On Noether’s bound for polynomial invariants of a ﬁnite grou p, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 5–7

work page 2001

[8] [8]

Gandini, Ideals of subspace arrangements , Ph.D

F. Gandini, Ideals of subspace arrangements , Ph.D. Thesis, University of Michigan, Ann Arbor, May 2019, ORCID iD: 0000-0002-2619-3555

work page 2019

[9] [9]

Grayson and Michael E

Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/

work page

[10] [10]

Kirkman and J

E. Kirkman and J. Kuzmanovich, Fixed subrings of Noetherian graded regular rings , J. Al- gebra 288 (2005), no. 2, 463–484

work page 2005

[11] [11]

Kirkman, J

E. Kirkman, J. Kuzmanovich, and J. Zhang, Invariants of (-1)-skew polynomial rings under permutation representations, Recent advances in representation theory, quantum groups , al- gebraic geometry, and related topics, 155–192, Contemp. Ma th., vol. 623, Amer. Math. Soc., Providence, RI, 2014

work page 2014

[12] [12]

Kirkman, I

E. Kirkman, I. M. Musson, and D. S. Passman, Noetherian down-up algebras , Proc. Amer. Math. Soc. 127 (1999), no. 11, 3161–3167

work page 1999

[13] [13]

Neusel, Degree bounds: An invitation to postmodern invariant theor y, Topology and its Applications 154 (2007), no

M. Neusel, Degree bounds: An invitation to postmodern invariant theor y, Topology and its Applications 154 (2007), no. 4, 792–814

work page 2007

[14] [14]

Noether, Der endlichkeitssatz der invarianten endlicher gruppen , Math

E. Noether, Der endlichkeitssatz der invarianten endlicher gruppen , Math. Ann 77 (1916), 89–92

work page 1916

[15] [15]

Sezer, Sharpening the generalized Noether bound in the invariant t heory of ﬁnite groups , J

M. Sezer, Sharpening the generalized Noether bound in the invariant t heory of ﬁnite groups , J. Algebra 254 (2002), no. 2, 252–263

work page 2002

[16] [16]

Symonds, On the Castelnuovo-Mumford regularity of rings of polynomi al invariants , An- nals of Math

P. Symonds, On the Castelnuovo-Mumford regularity of rings of polynomi al invariants , An- nals of Math. 174 (2011), 499–517. W ake Forest University, Department of Mathematics and Stati stics, P. O. Box 7388, Winston-Salem, North Carolina 27109 E-mail address : ferrarl@wfu.edu W ake Forest University, Department of Mathematics and Stati stics, P. O. Box ...

work page 2011