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Separating polynomial invariants over non-closed fields of finite abelian groups

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Pith's one-line read Over the rationals, degree-3 invariants separate every prime-order cyclic orbit pair.

desk verdict First known field-dependence of the separating Noether number, with a clean proof; just needs the polarization step's hypotheses spelled out. read the letter →

arxiv 2506.22889 v1 pith:YXOTAMQG submitted 2025-06-28 math.AC math.RT

Mátyás Domokos This is my paper
classification math.ACmath.RT MSC 13A5013P1520C1594A12
keywords polynomialinvariantsseparatingNoethernumberdegreeboundsfiniteabeliangroupscyclicregularrepresentationmultireferencealignment
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that for the cyclic group $C_p$ of prime order $p$, rational polynomial invariants of degree at most $3$ separate every pair of distinct $C_p$-orbits in the regular representation $\mathbb{Q}^p$, and it derives the same conclusion for every finite-dimensional representation of $C_p$ over $\mathbb{Q}$. In invariant-theory terms, the separating Noether number of $C_p$ over $\mathbb{Q}$ is $3$ for $p\ge 3$ and $2$ for $p=2$. This is the first known instance where the separating Noether number over an infinite field differs from its value over the algebraic closure, where it is $p$. The paper also generalizes a degree bound for separating invariants of finite abelian groups from algebraically closed base fields to non-closed fields such as $\mathbb{R}$ and $\mathbb{Q}$, and it illustrates the gain by computing $\beta^{\mathbb{R}}_{\mathrm{sep}}(C_3\times C_3)=3$. A direct motivation is orbit recovery and multireference alignment, where the numerical data are rational or real and low-degree invariants are algorithmically attractive.

What carries the argument

The load-bearing object is the block monoid $B(G)$ of a finite abelian group $G$: the additive monoid of integer-valued functions $s$ on $G$ whose product $\prod_{g\in G} g^{s(g)}$ equals the identity, i.e. product-one sequences. Lemma 3.1 proves that every element of $B(G)$ lies in the $\mathbb{Z}$-span of the short sequences $\delta_{1_G}$, $\delta_g+\delta_{g^{-1}}$, and $\delta_g+\delta_h+\delta_{(gh)^{-1}}$, all of length at most $3$. When the representation is diagonalized over a cyclotomic extension, invariant monomials are indexed by $B(\hat G)$, so this spanning statement is what allows degree-$3$ monomials to separate vectors whose character coordinates are all non-zero. For fields $\mathbb{F}$ that are not algebraically closed, Theorem 5.4 replaces the diagonal basis with the action of the Galois group $\Gamma$ on characters: condition ($*$) asks that for every $\Gamma$-stable subset $I\subseteq\hat G$, the monoid $B(\hat G)_I$ is contained in the $\mathbb{Z}$-span of the chosen low-degree elements of $B(\hat G)_I$.

What would settle it

Search for a separated pair in the regular representation $\mathbb{Q}^p$ of $C_p$ with coordinates in $\{-2,-1,0,1,2\}$, for $p=3$ and $p=5$: if two vectors with distinct cyclic-shift orbits take equal values on every $C_p$-orbit sum of monomials of degree at most $3$, then $\beta^{\mathbb{Q}}_{\mathrm{sep}}(C_p)=3$ is false. The same test on the $2$-dimensional irreducible rational representation for $p=3$ would falsify the claimed bound for arbitrary finite-dimensional representations.

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Extended reading notes

Core claim

The central claim is that separating behavior over $\mathbb{Q}$ is governed by the arithmetic of $p$-th roots of unity. In the regular representation $V=\mathbb{Q}^p$ of $C_p$, extending scalars to $K=\mathbb{Q}(\omega)$ diagonalizes the action, and a rational vector has all non-trivial character coordinates non-zero unless it lies on the fixed diagonal; this follows from the minimal polynomial $1+x+\cdots+x^{p-1}$ of $\omega$. For vectors with all non-trivial coordinates non-zero, orbit separation is equivalent to separation by the invariant monomials associated with the short product-one sequences $\delta_{1_{\hat G}}$, $\delta_\chi+\delta_{\chi^{-1}}$, and $\delta_\chi+\delta_\rho+\delta_{(\chi\rho)^{-1}}$, because Lemma 3.1 shows these sequences generate the full block monoid over the integers. The remaining fixed-diagonal case is handled by the degree-$1$ invariant $x_{1_{\hat G}}$. The non-closed-field generalization (Theorem 5.4) states a sufficient condition: a set of invariant monomials separates orbits in $\mathcal{F}(G,\mathbb{F})$ if, for every $\Gamma$-stable subset $I$ of characters, the block monoid restricted to $I$ is contained in the integer span of the chosen monomials restricted to $I$.

Load-bearing premise

The step from the regular representation of $C_p$ to every finite-dimensional rational representation rests on the cited polarization result that separating degree does not grow when taking direct sums of the regular representation; the paper's own proof establishes degree $3$ only for the regular representation directly.

Editorial extensions

If this is right

  • For every prime $p$, $\beta^{\mathbb{Q}}_{\mathrm{sep}}(C_p)=2$ when $p=2$ and $3$ when $p\ge 3$; in particular all finite-dimensional rational representations of $C_p$ are separated by degree-$3$ invariants.
  • Since $\beta^{\mathbb{C}}_{\mathrm{sep}}(C_p)=p$ and $\beta^{\mathbb{R}}_{\mathrm{sep}}(C_p)=p$, the rational result is the first example where the separating Noether number over an infinite field is smaller than over its algebraic closure.
  • Theorem 5.4 and Corollary 5.6 give a general sufficient condition for low-degree monomials to separate orbits over non-closed fields such as $\mathbb{R}$ and $\mathbb{Q}$, checkable from the block monoid and the Galois action on characters.
  • The computation $\beta^{\mathbb{R}}_{\mathrm{sep}}(C_3\times C_3)=3$ shows the bound can beat the complex-field bound $\beta^{\mathbb{C}}_{\mathrm{sep}}(C_3\times C_3)=4$ in a concrete real representation.
  • For the regular representation of $C_p$ over $\mathbb{Q}$, orbit recovery of rational signals from cyclic shifts can be done from invariants of degree at most $3$, rather than degree $p$ as over $\mathbb{C}$ or $\mathbb{R}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Galois-block-monoid mechanism suggests that other finite abelian groups whose cyclotomic splitting field has few Galois orbits may also have separating degree much smaller than their complex separating Noether number; computing $\beta^{\mathbb{Q}}_{\mathrm{sep}}(C_3\times C_3)$ would test this.
  • The cited polarization step could be probed directly: computing $\beta_{\mathrm{sep}}(C_p,\mathcal{F}(C_p,\mathbb{Q})^{\oplus 2})$ would tell whether the passage from the regular representation to all representations needs an extra theorem or follows from the regular representation alone.
  • For number fields between $\mathbb{Q}$ and $\mathbb{Q}(\omega_p)$, the same criterion may produce intermediate separating degrees, interpolating between the algebraic-closure value $p$ and the rational value $3$; the paper does not compute these.
  • In multireference alignment with rational data, degree-$3$ separating invariants translate into low-order correlation statistics, and a natural next question is their numerical conditioning under noise.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies separating polynomial invariants of finite abelian groups over fields that are not algebraically closed. Its headline result, Theorem 4.1, shows that for the regular representation of the cyclic group C_p over Q, any two distinct orbits can be separated by an invariant of degree at most 3. Corollary 4.2 then asserts beta^Q_sep(C_p)=3 for p>=3 and =2 for p=2, producing the first known example where the separating Noether number over an infinite field differs from that over the algebraic closure. Propositions 4.4 and 4.5 give negative results for C_4 and S_3. In Section 5 the paper generalizes a monomial-based criterion from the algebraically closed case to arbitrary non-closed base fields of non-modular characteristic, and applies it in Proposition 5.8 to compute beta^R_sep(C_3 x C_3)=3. Section 6 gives a concrete real representation illustrating the improvement with explicit invariant lists.

Significance. If the results are correct, Theorem 4.1 and Corollary 4.2 are a substantial contribution: they provide the first example of a separating Noether number over an infinite field differing from its value over the algebraic closure, and they sharpen the analysis of orbit recovery over Q, which is directly relevant to multireference alignment. The proof of Theorem 4.1 is elementary and largely self-contained, resting on a clean block-monoid argument and explicit character-basis computations; the negative results in Propositions 4.4 and 4.5 are useful boundary cases. The paper also gives a clear sufficient condition in Theorem 5.4 for separating orbits in the regular representation over non-closed fields, together with an explicit worked example in Section 6. The main weakness is that the step from the regular representation to arbitrary finite-dimensional representations in Corollary 4.2 is outsourced to citations whose hypotheses are not stated or verified, and the analogous step reappears with a caveat in Corollary 5.6(ii).

major comments (2)
  1. [Corollary 4.2] The reduction from the regular representation to all finite-dimensional Q[C_p]-modules is not made checkable. The proof bounds beta_sep(C_p, F(C_p,Q)) in Theorem 4.1, then asserts that beta^Q_sep(C_p) equals beta_sep(C_p, F(C_p,Q)^{⊕2}) via [11, Propositions 4.3 and 4.4] and that this equals beta_sep(C_p, F(C_p,Q)) via [14, Theorem 3.4]. Neither the hypotheses nor the statements of these cited results are given. This matters because the abstract's claim for every finite-dimensional representation of C_p rests entirely on this polarization step. The authors should either state the exact hypotheses of [14, Theorem 3.4] and [11, Propositions 4.3-4.4], verify that they hold for Q and for the regular representation, or provide a direct proof of the two-term reduction in this setting.
  2. [Corollary 5.6(ii)] A second occurrence of the same load-bearing reduction appears in the proof of Corollary 5.6(ii). The text says that a variant of Lemma 5.3, Lemma 5.5, and Theorem 5.4 'can be proved in the same way', and only alternatively cites [14, Theorem 3.4(ii)] with the caveat 'if F is large enough (say F is infinite)'. Since the conclusion beta^F_sep(G) <= d covers all finite-dimensional representations, this is exactly the type of polarization statement that must be either proved or precisely cited with hypotheses. The caveat is especially telling, as it suggests the cited theorem may carry conditions that are not spelled out in the main text. Please make this step fully explicit.
minor comments (5)
  1. [Section 2] There is a typographical error in the definition of the group action: 'for f ∈ F]V ]' should read 'for f ∈ F[V]'.
  2. [Proposition 4.5] In the proof, the notation 'x2 := x1x2 + x2x3 + x1x3' conflicts with the coordinate function x2; using s2 for the second elementary symmetric polynomial would avoid ambiguity.
  3. [Proposition 5.8] The classification of irreducible elements of B(C_3 x C_3) of length greater than 2 is only sketched. The sentence 'After these observations it is easy to sort out the few possibilities' is not a proof of the classification, and the subsequent argument depends on this classification. Please expand the enumeration or provide a verifiable computation.
  4. [Section 6] The explicit invariant lists are said to be computed with CoCalc, but no code or independent certificate is provided. For reproducibility, it would be helpful to include the underlying computation or a short verification script.
  5. [Proposition 5.1] The authors note that Proposition 5.1 'must be well known' but give no reference. Since the proof is supplied, this is only a presentation issue, but a reference would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main degree-3 separation theorem is derived self-containedly, and the cited polarization results are external prior theorems rather than re-imported conclusions.

full rationale

Theorem 4.1 is proved directly: it assumes that all rational invariants of degree at most 3 agree on v and w, extends scalars to K = Q(omega), and uses Lemma 3.1 and Proposition 3.2 to force Gv = Gw. The block-monoid argument is self-contained and does not presuppose the conclusion. Corollary 4.2's lower bound is an explicit orbit-pair computation, and its upper bound for arbitrary representations uses the standard direct-summand embedding into a direct sum of regular representations together with cited results [11] and [14]. These citations are external prior theorems, not definitions or fitted inputs; whether their hypotheses are satisfied is a verification/correctness matter, not a circular reduction. Section 5's Theorem 5.4 and Corollary 5.6 give a self-contained sufficient condition in terms of the block monoid. The only soft spot is that the polarization step in Corollary 4.2 and the 'variant can be proved in the same way' passage in Corollary 5.6 are not fully expanded, but neither passage makes a prediction equal to its own input by construction. No fitted parameter is renamed as a prediction, no ansatz is smuggled in via citation, and no uniqueness theorem is imported from the authors' prior work to forbid alternatives.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. The central results rest on standard invariant-theoretic separation, the non-modular hypothesis, cyclotomic Galois descent, and one cited polarization theorem; the only unproved external dependency is [14, Theorem 3.4(ii)].

assumptions (6)
  • standard math Finite group orbits are separated by polynomial invariants, Eq. (1).
    Used in Proposition 3.2 and Theorem 5.4 to conclude that equality of all invariants implies equality of orbits; cited to [10], [19], [14].
  • domain assumption The characteristic of F does not divide |G|, the non-modular hypothesis.
    Standing hypothesis throughout; needed for the Reynolds operator, base change of invariants, and decomposition into irreducible representations.
  • standard math K = F(omega) is a Galois extension of F with Galois group Gamma and fixed field F.
    Foundation of the Galois-descent Lemmas 5.2 and 5.3, where Gamma-invariance characterizes the original rational or real representation.
  • standard math Every finite-dimensional FG-module is a direct summand of a direct sum of copies of the regular representation.
    Used in Corollary 4.2 and Corollary 5.6 to reduce arbitrary representations to regular representations.
  • standard math [14, Theorem 3.4(ii)]: beta_sep(G, V direct sum m) = beta_sep(G, V) under suitable hypotheses, for example an infinite base field.
    Quoted in Corollary 4.2 and Corollary 5.6 to identify the maximum over m with the regular representation; not proven or stated fully in the paper.
  • standard math The minimal polynomial of a primitive p-th root of unity over Q is 1 + x + ... + x^(p-1).
    In Theorem 4.1 this forces the dichotomy that a vector with a zero non-trivial character coordinate is a multiple of the all-ones vector.

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Pith. "Pith review of Separating polynomial invariants over non-closed fields of finite abelian groups." pith.science (2026). https://pith.science/paper/YXOTAMQG

@misc{pith2026250622889,
  author       = {Pith},
  title        = {Pith review of: Separating polynomial invariants over non-closed fields of finite abelian groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YXOTAMQG}},
  note         = {Machine review of arXiv:2506.22889}
}
abstract

It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating invariants for representations of finite abelian groups over algebraically closed base fields of non-modular characteristic is generalized for the case of base fields that are not algebraically closed (like the fields of real or rational numbers).

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Works this paper leans on

30 extracted references · 28 canonical work pages

  1. [12]

    Domokos, Degree bound for separating invariants of finite abelian groups, Proc

    M. Domokos, Degree bound for separating invariants of finite abelian groups, Proc. Amer. Math. Soc. 145 (2017), 3695-3708

    work page 2017

  2. [11]

    Domokos, Typical separating invariants, Transform

    M. Domokos, Typical separating invariants, Transform. Groups 12 (2007), 49-63

    work page 2007

  3. [14]

    Draisma, G

    J. Draisma, G. Kemper, D. Wehlau, Polarization of separating invariants, Canad. J. Math. 60 (2008) 556-571

    work page 2008

  4. [1]

    A. S. Bandeira, B. Blum-Smith, J. Kileel, J. Niles-Weed, A. Perry, A. S. Wein, Estimation under group actions: recovering orbits from invariants, Appl. Comput. Harmon. Anal. 66 (2023), 236-319

    work page 2023

  5. [2]

    A. S. Bandeira, J. Niles-Weed, Ph. Rigollet, Optimal rates of estimation for multi-reference alignment, Math. Stat. Learn. 2 (2019), 25-75

    work page 2019

  6. [3]

    Blum-Smith, H

    B. Blum-Smith, H. Derksen, Generic orbits, normal bases, and generation degree for fields of rational invariants, arXiv:2506.05650

    arXiv

  7. [4]

    Blum-Smith, T

    B. Blum-Smith, T. Garcia, R. Hidalgo, C. Rodriguez, Degree bounds for fields of rational invariants of Z/pZ and other finite groups, J. Pure Appl. Algebra 228 (2024), Article ID 107693

    work page 2024

  8. [5]

    Cahill, A

    J. Cahill, A. Contreras, A. C. Hip, Stable separation of orbits for finite abelian group actions, J. Fourier Anal. Appl. 30 (2024), Paper No. 12, 16 p

    work page 2024

Show all 30 references
  1. [6]

    Cahill, J

    J. Cahill, J. W. Iverson, D. G. Mixon, Towards a bilipschitz invariant theory, Appl. Comput. Harmon. Anal. 72 (2024), Article ID 101669, 27 p

    2024

  2. [7]

    Cahill, J

    J. Cahill, J. W. Iverson, D. G. Mixon, D. Packer, Group-invariant max filtering, Foundations of Computational Mathematics 25 (2025), 1047-1084

    2025

  3. [8]

    Cziszter, M

    K. Cziszter, M. Domokos, Lower bounds on the Noether number, Transform. Groups 24 (2019), 823-834

    2019

  4. [9]

    Cziszter, M

    K. Cziszter, M. Domokos, I. Sz¨ oll˝ osi, The Noether numbers and the Davenport constants of the groups of order less than 32, J. Algebra 510 (2018), 513-541

    2018

  5. [10]

    Derksen, G

    H. Derksen, G. Kemper, Computational Invariant Theory, Second Edition, Encyclopaedia of Math- ematical Sciences 130, Invariant Theory of Algebraic Transformation Groups VIII, Springer-Verlag, Berlin, Heidelberg, 2015

    2015

  6. [13]

    Domokos, Separating monomials for diagonalizable actions, Bull

    M. Domokos, Separating monomials for diagonalizable actions, Bull. Lond. Math. Soc. 55 (2023), 205-223

    2023

  7. [15]

    Dufresne, J

    E. Dufresne, J. Jeffries, Mapping toric varieties into low dimensional spaces, Trans. Amer. Math. Soc., to appear; https://doi.org/10.1090/tran/7026, arXiv:1602.07585v2

    arXiv

  8. [16]

    Edidin, J

    D. Edidin, J. Katz, Orbit recovery from invariants of very low degree, arXiv:2503.00009

    arXiv

  9. [17]

    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), 23-32

    2000

  10. [18]

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

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

    2001

  11. [19]

    Kemper, Separating invariants, J

    G. Kemper, Separating invariants, J. Symb. Comput. 44 (2009),1212-1222

    2009

  12. [20]

    Kohls, H

    M. Kohls, H. Kraft, Degree bounds for separating invariants, Math. Res. Lett. 17 (2010), 1171-1182

    2010

  13. [21]

    Narkiewicz, Finite abelian groups and factorization problems, Colloq

    W. Narkiewicz, Finite abelian groups and factorization problems, Colloq. Math. 17 (1979), 319-330

    1979

  14. [22]

    Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math

    E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77 (1916), 89-92. SEPARATING POLYNOMIAL INV ARIANTS OVER NON-CLOSED FIELDS OF FINITE ABELIAN GROUPS 16

    1916

  15. [23]

    Perry, J

    A. Perry, J. Weed, A. S. Bandeira, Ph. Rigollet, A. Singer, The sample complexity of multiref-erence alignment, SIAM J. Math. Data Sci. 1(3) (2019) 497-517

    2019

  16. [24]

    D. R. Richman, Invariants of finite groups over fields of characteristic p, Adv. Math. 124 (1996), 25-48

    1996

  17. [25]

    Sagemath, Inc., CoCalc – Collaborative Calculation and Data Science, 2020, https://cocalc.com

    2020

  18. [26]

    Schefler, The separating Noether number of the direct sum of several copies of a cyclic group, Proc

    B. Schefler, The separating Noether number of the direct sum of several copies of a cyclic group, Proc. Amer. Math. Soc. 153 (2025), 69-79

    2025

  19. [27]

    Schefler, The separating Noether number of abelian groups of rank two, J

    B. Schefler, The separating Noether number of abelian groups of rank two, J. Comb. Theory, Ser. A 209 (2025), Paper no. 105951

    2025

  20. [28]

    Schefler, K

    B. Schefler, K. Zhao, Q. Zhong, On the separating Noether number of finite abelian groups, arXiv:2503.01296v2

    arXiv

  21. [29]

    Topics in invariant theory

    B. J. Schmid, Finite groups and invariant theory, in “Topics in invariant theory” (M.-P. Malliavin, ed.), Lecture notes in mathematics, no. 1478, Springer, 1989-90, pp. 35-66

    1989

  22. [30]

    Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer Verlag, New York, 1977

    J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer Verlag, New York, 1977. HUN-REN Alfr ´ed R ´enyi Institute of Mathematics, Re ´altanoda utca 13-15, 1053 Bu- dapest, Hungary, ORCID iD: https://orcid.org/0000-0002-0189-8831 Email ...

    1977

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