pith. sign in

arxiv: 2606.04346 · v1 · pith:PB5J72W5new · submitted 2026-06-03 · 🧮 math.AC

Hilbert-Kunz multiplicity of quadrics decreases

Cheng Meng This is my paper

Pith reviewed 2026-06-28 03:44 UTC · model grok-4.3

classification 🧮 math.AC
keywords Green ringsHilbert-Kunz multiplicityFermat quadricsp-adic expansionsGelfand transformYoshida conjecturetensor productscharacteristic p
0
0 comments X

The pith

The Hilbert-Kunz multiplicity of Fermat quadrics decreases as the characteristic increases.

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

The paper proves that the Green ring of Z/p^e Z is the e-fold tensor product of the Green ring of Z/p Z, with the isomorphism given explicitly by p-adic expansions. This structural result is used to calculate the Hilbert-Kunz function and multiplicity for Fermat quadrics in any characteristic. An analytical expression for the multiplicity is then derived using the Gelfand transform on the Green ring. The expression demonstrates that the multiplicity is strictly decreasing in the prime characteristic, thereby confirming a conjecture of Yoshida.

Core claim

The Green ring of Z/p^e Z is isomorphic to the e-fold tensor product of copies of the Green ring of Z/p Z via the map induced by p-adic digit expansions. Consequently the Hilbert-Kunz multiplicity of a Fermat quadric hypersurface admits a closed-form analytic expression obtained by applying the Gelfand transform to this ring; the resulting function of the characteristic is monotonically decreasing.

What carries the argument

The isomorphism of Green rings induced by p-adic expansions, which reduces computations in p-power characteristics to the prime case and enables the Gelfand transform to produce an explicit formula for the Hilbert-Kunz multiplicity.

If this is right

  • Explicit formulas for Hilbert-Kunz multiplicities of Fermat quadrics become available in every characteristic.
  • The multiplicity decreases monotonically with increasing prime characteristic.
  • Yoshida's conjecture on the behavior of these multiplicities is resolved affirmatively.
  • The tensor product decomposition determines the ring structure completely from the prime case.

Where Pith is reading between the lines

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

  • The same tensor product decomposition may apply to other representation rings or invariants in positive characteristic.
  • The monotonicity could be used to obtain asymptotic bounds on the multiplicity as the characteristic tends to infinity.
  • Analogous results might hold for Fermat hypersurfaces of higher degree if similar ring structures can be identified.

Load-bearing premise

That the Gelfand transform of the Green ring produces an analytical expression for the Hilbert-Kunz multiplicity which permits direct comparison between different characteristics.

What would settle it

Computing the Hilbert-Kunz multiplicity numerically for the Fermat quadric equation in characteristics 2, 3 and 5 and verifying that the values form a strictly decreasing sequence.

read the original abstract

In this paper we prove that the Green ring of $\mathbb{Z}/p^e\mathbb{Z}$ is the $e$-fold tensor product of the Green ring of $\mathbb{Z}/p\mathbb{Z}$, and this isomorphism is given by $p$-adic expansions of integers. As an application of this isomorphism, we compute the Hilbert-Kunz function and Hilbert-Kunz multiplicity of Fermat quadrics. Then we use Gelfand transform of the Green ring to give an analytical expression of this Hilbert-Kunz multiplicity, and prove that it decreases with its characteristic, thus giving a positive answer to a conjecture of Yoshida.

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 / 2 minor

Summary. The paper proves that the Green ring of ℤ/p^eℤ is isomorphic to the e-fold tensor product of the Green ring of ℤ/pℤ, with the isomorphism given explicitly by p-adic expansions. As an application it computes the Hilbert-Kunz function and multiplicity of Fermat quadrics over these rings, obtains a closed-form analytical expression for the multiplicity via the Gelfand transform of the Green ring, and shows that the multiplicity is strictly decreasing in the characteristic p, thereby confirming Yoshida's conjecture.

Significance. If the isomorphism and the monotonicity statement hold, the work supplies both a structural result on Green rings that may be of independent interest and the first positive resolution of Yoshida's conjecture for the Fermat quadric case. The use of the Gelfand transform to produce an explicit, comparable expression across characteristics is a novel technical device in this setting.

major comments (2)
  1. [Application section (after the isomorphism theorem)] The central application (reduction of HK multiplicity of Fermat quadrics to the e=1 case via the tensor-product isomorphism, followed by the Gelfand-transform expression) is load-bearing for the claimed decrease in p. The manuscript must supply an explicit verification that the spectrum and the idempotents appearing in the Gelfand transform introduce no additional p-dependent normalizations or convergence factors that could interfere with monotonicity; the abstract alone does not exhibit this check.
  2. [Gelfand-transform paragraph and subsequent comparison] The proof that the analytical expression obtained from the Gelfand transform decreases monotonically with p must be written out in full; it is not immediate from the ring isomorphism that the p-dependence enters solely through the base ring.
minor comments (2)
  1. Notation for the Green ring and its spectrum should be introduced once and used consistently; several passages reuse symbols without redefinition.
  2. The statement of the main isomorphism theorem would benefit from an explicit description of the generators and relations on both sides.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: [Application section (after the isomorphism theorem)] The central application (reduction of HK multiplicity of Fermat quadrics to the e=1 case via the tensor-product isomorphism, followed by the Gelfand-transform expression) is load-bearing for the claimed decrease in p. The manuscript must supply an explicit verification that the spectrum and the idempotents appearing in the Gelfand transform introduce no additional p-dependent normalizations or convergence factors that could interfere with monotonicity; the abstract alone does not exhibit this check.

    Authors: We agree that an explicit verification strengthens the argument. In the revised manuscript we will insert a short subsection immediately after the isomorphism theorem that computes the spectrum of the Green ring (as the set of ring homomorphisms to ℂ) and lists the idempotents explicitly via the p-adic decomposition. We will verify that these data are independent of the prime p and introduce no extra normalization or convergence factors; the only p-dependence therefore flows through the base ring ℤ/pℤ and the already-established e=1 Hilbert-Kunz multiplicity formula. revision: yes

  2. Referee: [Gelfand-transform paragraph and subsequent comparison] The proof that the analytical expression obtained from the Gelfand transform decreases monotonically with p must be written out in full; it is not immediate from the ring isomorphism that the p-dependence enters solely through the base ring.

    Authors: We acknowledge that the monotonicity argument, while present in outline, benefits from a fully expanded write-up. In the revision we will expand the Gelfand-transform paragraph into a self-contained proof that first recalls the explicit closed-form expression obtained from the transform, then isolates the p-dependence to the e=1 case (via the known formula for Fermat quadrics over ℤ/pℤ), and finally compares the resulting rational functions for successive primes p to establish strict decrease. This makes transparent that no hidden p-dependent factors arise from the isomorphism itself. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on explicit ring isomorphism and external Gelfand transform application

full rationale

The paper's core steps are (1) proving the Green ring of Z/p^e Z equals the e-fold tensor product of the Green ring of Z/p Z via an explicit p-adic expansion map, and (2) applying the Gelfand transform to obtain a closed-form HK multiplicity expression whose monotonicity in p is then verified directly. Neither step reduces to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation whose content is merely renamed. The isomorphism is constructed and verified independently of the target HK multiplicity; the Gelfand transform is a standard functional-analytic tool applied after the ring structure is established. The decrease claim is presented as a consequence of comparing the resulting analytic expressions across primes, not as an input. No equations are shown to be tautological by construction, and the derivation chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, ad-hoc axioms, or invented entities are mentioned. The work relies on standard background definitions of Green rings, tensor products, Gelfand transforms, and Hilbert-Kunz multiplicity.

axioms (2)
  • standard math Standard algebraic properties of Green rings and their tensor products over Z/p^e Z
    Invoked implicitly to establish the claimed isomorphism.
  • standard math Existence and basic properties of the Gelfand transform on the Green ring
    Used to obtain the analytical expression for the multiplicity.

pith-pipeline@v0.9.1-grok · 5612 in / 1358 out tokens · 43630 ms · 2026-06-28T03:44:24.548649+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

25 extracted references · 2 linked inside Pith

  1. [1]

    Aberbach and Florian Enescu

    Ian M. Aberbach and Florian Enescu. Lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension. volume 57, pages 1–16. 2008. Special volume in honor of Melvin Hochster

    2008

  2. [2]

    Aberbach and Florian Enescu

    Ian M. Aberbach and Florian Enescu. New estimates of Hilbert-Kunz multiplicities for local rings of fixed dimension.Nagoya Math. J., 212:59–85, 2013

    2013

  3. [3]

    David J. Benson. Modular representation theory and commutative Banach algebras.Mem. Amer. Math. Soc., 298(1488):v+118, 2024

    2024

  4. [4]

    Irrational Hilbert-Kunz multiplicities.arXiv e-prints, page arXiv:1305.5873, May 2013

    Holger Brenner. Irrational Hilbert-Kunz multiplicities.arXiv e-prints, page arXiv:1305.5873, May 2013

    Pith/arXiv arXiv 2013

  5. [5]

    Strong Watanabe-Yoshida conjecture for complete intersections.arXiv e-prints, page arXiv:2506.09019, June 2025

    Joel Castillo-Rey. Strong Watanabe-Yoshida conjecture for complete intersections.arXiv e-prints, page arXiv:2506.09019, June 2025

    arXiv 2025

  6. [6]

    Cox-Steib and Ian M

    Nicholas O. Cox-Steib and Ian M. Aberbach. Bounds for the Hilbert-Kunz multiplicity of singular rings.Acta Math. Vietnam., 49(1):39–60, 2024

    2024

  7. [7]

    On the upper semi-continuity of the Hilbert-Kunz multi- plicity.J

    Florian Enescu and Kazuma Shimomoto. On the upper semi-continuity of the Hilbert-Kunz multi- plicity.J. Algebra, 285(1):222–237, 2005

    2005

  8. [8]

    J. A. Green. The modular representation algebra of a finite group.Illinois J. Math., 6:607–619, 1962

    1962

  9. [9]

    Han and P

    C. Han and P. Monsky. Some surprising Hilbert-Kunz functions.Math. Z., 214(1):119–135, 1993

    1993

  10. [10]

    Hilbert-Kunz multiplicity and the F-signature

    Craig Huneke. Hilbert-Kunz multiplicity and the F-signature. InCommutative algebra, pages 485–

  11. [11]

    Springer, New York, 2013

    2013

  12. [12]

    Ken ichi Yoshida. Small Hilbert-Kunz multiplicity and (A1)-type singularity.Proceedings of the 4thJapan-Vietnam Joint Seminar on Commutative Algebra by and for Young Mathematicians, Meiji University, 2019

    2019

  13. [13]

    A Proof of a conjecture of Watanabe–Yoshida via Ehrhart Theory.arXiv e-prints, page arXiv:2512.20442, December 2025

    Yakob Kahane. A Proof of a conjecture of Watanabe–Yoshida via Ehrhart Theory.arXiv e-prints, page arXiv:2512.20442, December 2025

    arXiv 2025

  14. [14]

    Analysis in Hilbert-Kunz theory.arXiv e-prints, page arXiv:2507.13898, July 2025

    Cheng Meng. Analysis in Hilbert-Kunz theory.arXiv e-prints, page arXiv:2507.13898, July 2025

    arXiv 2025

  15. [15]

    Asymptotic behavior of modular representations over abelianp-groups.arXiv e-prints, page arXiv:2603.11592, March 2026

    Cheng Meng. Asymptotic behavior of modular representations over abelianp-groups.arXiv e-prints, page arXiv:2603.11592, March 2026

    Pith/arXiv arXiv 2026

  16. [16]

    P. Monsky. The Hilbert-Kunz function.Math. Ann., 263(1):43–49, 1983

    1983

  17. [17]

    Rationality of Hilbert-Kunz multiplicities: a likely counterexample

    Paul Monsky. Rationality of Hilbert-Kunz multiplicities: a likely counterexample. volume 57, pages 605–613. 2008. Special volume in honor of Melvin Hochster

    2008

  18. [18]

    Paul Monsky and Pedro Teixeira.p-fractals and power series. II. Some applications to Hilbert-Kunz theory.J. Algebra, 304(1):237–255, 2006

    2006

  19. [19]

    Hilbert-Kunz multiplicity of quadrics via Ehrhart theory.arXiv e-prints, page arXiv:2508.17915, August 2025

    Igor Pak, Boris Shapiro, Ilya Smirnov, and Ken-ichi Yoshida. Hilbert-Kunz multiplicity of quadrics via Ehrhart theory.arXiv e-prints, page arXiv:2508.17915, August 2025

    arXiv 2025

  20. [20]

    J.-C. Renaud. The decomposition of products in the modular representation ring of a cyclic group of prime power order.J. Algebra, 58(1):1–11, 1979

    1979

  21. [21]

    The modular representation ring of a cyclicp-group.Proc

    Bhama Srinivasan. The modular representation ring of a cyclicp-group.Proc. London Math. Soc. (3), 14:677–688, 1964

    1964

  22. [22]

    The Hilbert-Kunz density functions of quadric hypersurfaces.Adv

    Vijaylaxmi Trivedi. The Hilbert-Kunz density functions of quadric hypersurfaces.Adv. Math., 430:Paper No. 109207, 63, 2023

    2023

  23. [23]

    Hilbert-Kunz multiplicity and an inequality between mul- tiplicity and colength.J

    Kei-ichi Watanabe and Ken-ichi Yoshida. Hilbert-Kunz multiplicity and an inequality between mul- tiplicity and colength.J. Algebra, 230(1):295–317, 2000

    2000

  24. [24]

    Hilbert-Kunz multiplicity of two-dimensional local rings

    Kei-Ichi Watanabe and Ken-Ichi Yoshida. Hilbert-Kunz multiplicity of two-dimensional local rings. Nagoya Math. J., 162:87–110, 2001

    2001

  25. [25]

    Hilbert-Kunz multiplicity of three-dimensional local rings

    Kei-ichi Watanabe and Ken-ichi Yoshida. Hilbert-Kunz multiplicity of three-dimensional local rings. Nagoya Math. J., 177:47–75, 2005. Cheng Meng, Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China.Email:cheng319000@mail.tsinghua.edu.cn

    2005