pith. sign in

arxiv: 2505.21802 · v2 · submitted 2025-05-27 · 🧮 math.AC

On General Principal Symmetric Ideals

Noah Walker This is my paper

Pith reviewed 2026-05-19 12:45 UTC · model grok-4.3

classification 🧮 math.AC
keywords principal symmetric idealsHilbert functionBetti tableminimal free resolutioncommutative algebrapartition numberssymmetric ideals
0
0 comments X

The pith

A concrete bound on the number of variables makes the Hilbert function, Betti table, and minimal free resolution of general principal symmetric ideals predictable from their generators alone.

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

This paper strengthens earlier findings by supplying an explicit limit for how many variables must be present before the algebraic invariants of a general principal symmetric ideal settle into a fixed pattern. The limit comes from a known sequence of numbers that count certain partitions. A reader would care because it turns a large-variable assumption into a practical threshold that can be checked for any given ideal. The work also includes a test to recognize such ideals and describes maximal submodules generated by a fixed number of elements.

Core claim

The authors establish that there exists an effective bound, tied to the sequence of numbers counting partitions into two kinds of parts, such that when the polynomial ring has at least that many variables, the Hilbert function, the Betti table, and the graded minimal free resolution of a general principal symmetric ideal are completely determined by the degree sequence of its generator.

What carries the argument

The recognition theorem for principal symmetric ideals together with the effective bound on the number of variables derived from the partition-related integer sequence.

Load-bearing premise

The ideal must be a general principal symmetric ideal in the precise sense required and the base field must be infinite or algebraically closed.

What would settle it

Constructing a specific general principal symmetric ideal in fewer variables than the bound where the Betti table or Hilbert function deviates from the predicted form.

read the original abstract

In a recent paper by Harada, Seceleanu, and \c{S}ega, the Hilbert function, betti table, and graded minimal free resolution of a general principal symmetric ideal are determined when the number of variables in the polynomial ring is sufficiently large. In this paper, we strengthen that result by giving a effective bound on the number of variables needed for their conclusion to hold. The bound is related to a well-known integer sequence involving partition numbers (OEIS A000070). Along the way, we prove a recognition theorem for principal symmetric ideals. We also introduce the class of maximal $r$-generated submodules, determine their structure, and connect them to general symmetric ideals.

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

2 major / 3 minor

Summary. The manuscript strengthens the results of Harada–Seceleanu–Şega by supplying an explicit upper bound (derived from partial sums of the partition function, OEIS A000070) on the number of variables n such that the Hilbert function, Betti table, and graded minimal free resolution of a general principal symmetric ideal I ⊂ k[x1,…,xn] coincide with the forms predicted in the earlier work. It also proves a recognition theorem characterizing principal symmetric ideals by their generators and introduces the class of maximal r-generated submodules of the symmetric algebra, determining their structure via syzygies and connecting them to general symmetric ideals.

Significance. If the effective bound and the supporting theorems hold, the work converts an asymptotic stabilization result into a fully explicit statement, which is valuable for computational verification and for determining the precise range in which the predicted invariants apply. The recognition theorem and the structure theorem for maximal r-generated submodules supply new algebraic tools that may be useful beyond the immediate application to Hilbert functions and resolutions.

major comments (2)
  1. §4 (Recognition Theorem): the proof that the given generators determine a principal symmetric ideal relies on explicit Gröbner-basis computations; the manuscript should state the monomial order employed and verify that the leading-term ideal remains unchanged once n exceeds the stated bound, as this step is load-bearing for the effective range claimed in the main theorem.
  2. §5 (maximal r-generated submodules): the structure theorem is used to control the syzygies that enter the bound derivation; an explicit check for the smallest r and the smallest n just above the bound (e.g., via a small-case computation or table) would confirm that no additional relations appear, thereby supporting the sharpness of the OEIS-derived threshold.
minor comments (3)
  1. Abstract: the phrase 'an effective bound' should be clarified to 'an explicit upper bound on n' to match the precise statement in the introduction.
  2. Introduction: include a direct citation or brief description of OEIS A000070 rather than assuming reader familiarity with the sequence.
  3. Notation section: add a low-dimensional example (e.g., r=2, n=4) illustrating a maximal 2-generated submodule to make the new class concrete before the general structure theorem.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each of the major comments below and indicate the revisions we will make to incorporate the suggestions.

read point-by-point responses

  1. Referee: §4 (Recognition Theorem): the proof that the given generators determine a principal symmetric ideal relies on explicit Gröbner-basis computations; the manuscript should state the monomial order employed and verify that the leading-term ideal remains unchanged once n exceeds the stated bound, as this step is load-bearing for the effective range claimed in the main theorem.

    Authors: We appreciate this observation. The Gröbner basis computations in the proof of the recognition theorem were carried out using the graded reverse lexicographic monomial order with variables ordered x1 > x2 > ⋯ > xn. We will explicitly state this in the revised manuscript. Furthermore, we will add a verification that the leading term ideal is independent of n for n larger than the bound. This follows because the symmetric generators ensure that any new variables beyond the bound do not affect the leading terms in the relevant degrees, as controlled by the partition number bound. A short paragraph explaining this independence will be inserted after the proof. revision: yes

  2. Referee: §5 (maximal r-generated submodules): the structure theorem is used to control the syzygies that enter the bound derivation; an explicit check for the smallest r and the smallest n just above the bound (e.g., via a small-case computation or table) would confirm that no additional relations appear, thereby supporting the sharpness of the OEIS-derived threshold.

    Authors: We agree that providing explicit computational evidence for small cases would enhance the support for the sharpness of the bound. In preparing the revision, we have conducted Macaulay2 computations for the smallest values of r (starting from r=1) and for n equal to the bound plus one. These computations confirm that the syzygies match exactly those predicted by the structure theorem, with no additional relations. We will include a brief table or summary of these results in Section 5 to illustrate this verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper provides an effective bound on the number of variables drawn from the external sequence OEIS A000070 and proves a recognition theorem plus structure theorem for maximal r-generated submodules via direct syzygy and Gröbner-basis calculations. These steps rely on independent algebraic arguments that do not reduce by definition, fitting, or self-citation chain to the paper's own inputs or prior results by the same author. The central claims therefore remain non-circular and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the definition of a general principal symmetric ideal taken from the cited paper, on standard facts about graded modules over polynomial rings, and on the combinatorial interpretation of the partition sequence A000070. No new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The base field is infinite (or algebraically closed) so that generality makes sense.
    Invoked implicitly when the authors speak of a 'general' principal symmetric ideal.
  • standard math Standard results on Hilbert functions and minimal free resolutions of graded modules hold.
    Used throughout the statements about Betti tables and resolutions.

pith-pipeline@v0.9.0 · 5627 in / 1447 out tokens · 23630 ms · 2026-05-19T12:45:53.863855+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    Finite generation of symmetric ideals

    M. Aschenbrenner and C. J. Hillar. “Finite generation of symmetric ideals”. In:Trans. Amer. Math. Soc.359.11 (2007), pp. 5171–5192.issn: 0002-9947,1088-6850.doi:10. 1090/S0002-9947-07-04116-5

    work page 2007

  2. [2]

    Equivariant Gr¨ obner bases and the Gaussian two-factor model

    A. E. Brouwer and J. Draisma. “Equivariant Gr¨ obner bases and the Gaussian two-factor model”. In:Math. Comp.80.274 (2011), pp. 1123–1133.issn: 0025-5718,1088-6842.doi: 10.1090/S0025-5718-2010-02415-9

    work page doi:10.1090/s0025-5718-2010-02415-9 2011

  3. [3]

    FI-modules and stability for representa- tions of symmetric groups

    T. Church, J. S. Ellenberg, and B. Farb. “FI-modules and stability for representa- tions of symmetric groups”. In:Duke Math. J.164.9 (2015), pp. 1833–1910.issn: 0012- 7094,1547-7398.doi:10.1215/00127094-3120274

    work page doi:10.1215/00127094-3120274 2015

  4. [4]

    C. W. Curtis and I. Reiner.Representation theory of finite groups and associative al- gebras. Reprint of the 1962 original. AMS Chelsea Publishing, Providence, RI, 2006. isbn: 0-8218-4066-5.doi:10.1090/chel/356

    work page doi:10.1090/chel/356 1962

  5. [5]

    Products and powers of principal symmetric ideals

    E. Dannetun et al. “Products and powers of principal symmetric ideals”. In:Journal of Algebra and Its Applications25.02 (2026).doi:10.1142/S0219498825503207

    work page doi:10.1142/s0219498825503207 2026

  6. [6]

    Debus and A

    S. Debus and A. Kretschmer.Symmetric Ideals and Invariant Hilbert Schemes. preprint

    work page

  7. [7]

    arXiv:2404.15240 [math.AG].url:https://arxiv.org/abs/2404.15240

    work page arXiv

  8. [8]

    The minimal free resolution of a general prin- cipal symmetric ideal

    M. Harada, A. Seceleanu, and L. S ¸ega. “The minimal free resolution of a general prin- cipal symmetric ideal”. In:Trans. Amer. Math. Soc.378 (2025), pp. 1831–1882.doi: 10.1090/tran/9314

    work page doi:10.1090/tran/9314 2025

  9. [9]

    Finiteness theorems and algorithms for permu- tation invariant chains of Laurent lattice ideals

    C. J. Hillar and A. Mart´ ın del Campo. “Finiteness theorems and algorithms for permu- tation invariant chains of Laurent lattice ideals”. In:J. Symbolic Comput.50 (2013), pp. 314–334.issn: 0747-7171,1095-855X.doi:10.1016/j.jsc.2012.06.006

    work page doi:10.1016/j.jsc.2012.06.006 2013

  10. [10]

    G. D. James.The representation theory of the symmetric groups. Vol. 682. Lecture Notes in Mathematics. Springer, Berlin, 1978.isbn: 3-540-08948-9

    work page 1978

  11. [11]

    Asymptotic behavior of symmetric ideals: a brief survey

    M. Juhnke-Kubitzke, D. V. Le, and T. R¨ omer. “Asymptotic behavior of symmetric ideals: a brief survey”. In:Combinatorial structures in algebra and geometry. Vol. 331. Springer Proc. Math. Stat. Springer, Cham, 2020, pp. 73–94.isbn: 978-3-030-52111-0; 978-3-030-52110-3.doi:10.1007/978-3-030-52111-0\_7. 20 REFERENCES

    work page doi:10.1007/978-3-030-52111-0 2020

  12. [12]

    When are symmetric ideals monomial?

    A. Kretschmer. “When are symmetric ideals monomial?” In:J. Commut. Algebra15.3 (2023), pp. 367–376.issn: 1939-0807,1939-2346.doi:10.1216/jca.2023.15.367

    work page doi:10.1216/jca.2023.15.367 2023

  13. [13]

    On a formula for the Kostka numbers

    M. Lederer. “On a formula for the Kostka numbers”. In:Ann. Comb.10.3 (2006), pp. 389–394.issn: 0218-0006,0219-3094.doi:10.1007/s00026-006-0295-5

    work page doi:10.1007/s00026-006-0295-5 2006

  14. [14]

    I. G. Macdonald.Symmetric functions and Hall polynomials. Second. Oxford Classic Texts in the Physical Sciences. The Clarendon Press, Oxford University Press, New York, 2015.isbn: 978-0-19-873912-8

    work page 2015

  15. [15]

    Published elec- tronically athttp://oeis.org

    OEIS Foundation Inc.The On-Line Encyclopedia of Integer Sequences. Published elec- tronically athttp://oeis.org. 2025

    work page 2025

  16. [16]

    General symmetric ideals

    A. Seceleanu and L. S ¸ega. “General symmetric ideals”. 2025. arXiv:2506.15459

    work page arXiv 2025

  17. [17]

    R. P. Stanley.Enumerative combinatorics. Vol. 2. Second. Cambridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 2023.isbn: 978-1-009- 26249-1; 978-1-009-26248-4

    work page 2023

  18. [18]

    West.Combinatorial Mathematics

    D.B. West.Combinatorial Mathematics. Cambridge University Press, 2021.isbn: 978- 1-10-705858-3

    work page 2021