A Hilbert-Kunz function with a periodic term that has a given period

Robin Baidya

arxiv: 1906.08821 · v1 · pith:6M7WRZ4Anew · submitted 2019-06-20 · 🧮 math.AC

A Hilbert-Kunz function with a periodic term that has a given period

Robin Baidya This is my paper

Pith reviewed 2026-05-25 18:44 UTC · model grok-4.3

classification 🧮 math.AC

keywords Hilbert-Kunz functionperiodic termone-dimensional local ringprime characteristiceventual periodicityMonsky

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The pith

For every positive integer π there exists a one-dimensional local ring of prime characteristic whose Hilbert-Kunz function has an immediately periodic term with period π.

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

Monsky proved that the Hilbert-Kunz function of any one-dimensional local ring in prime characteristic eventually decomposes into a linear term plus an eventually periodic term φ. Known examples had φ immediately periodic with period 1 or 2. The paper constructs rings in which φ is immediately periodic with any prescribed period π. This shows that the possible periods are unbounded. A reader cares because the construction demonstrates that the eventual periodicity result is sharp in the sense that immediate periodicity with arbitrary period is attainable.

Core claim

Monsky states that the Hilbert-Kunz function of a one-dimensional local ring of prime characteristic has a term φ that is eventually periodic. The paper shows that for every positive integer π there exists such a ring for which φ is immediately periodic with period π.

What carries the argument

The eventually periodic term φ appearing in the Hilbert-Kunz function of one-dimensional local rings in prime characteristic.

Load-bearing premise

Suitable one-dimensional local rings in prime characteristic exist whose periodic term φ can be made immediately periodic with any chosen period.

What would settle it

An explicit computation or proof that no one-dimensional local ring exists for which the periodic term φ has immediate period 3.

read the original abstract

A result of Monsky states that the Hilbert-Kunz function of a one-dimensional local ring of prime characteristic has a term $\phi$ that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, $\phi$ is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz and Monsky, $\phi$ is immediately periodic with period 2. We show that, for every positive integer $\pi$, there exists a ring for which $\phi$ is immediately periodic with period $\pi$.

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

The paper gives an explicit construction of one-dimensional local rings where the Hilbert-Kunz periodic term has any prescribed immediate period.

read the letter

The core result is a construction that lets you prescribe any positive integer as the immediate period of the Monsky periodic term in the Hilbert-Kunz function of a one-dimensional local ring in prime characteristic. They move past the known period-1 and period-2 cases by producing, for each π, a ring R_π whose associated function φ is immediately periodic with that exact period. The argument rests on direct length computations rather than existence arguments or extra conditions on π. That part is clean and parameter-free, which is the main thing the paper contributes. The verification that periodicity begins at the earliest possible index is also checked explicitly, so the claim matches what is shown. The limitation is scope: this stays inside the narrow setting of Hilbert-Kunz functions for one-dimensional rings and does not appear to affect wider questions in commutative algebra or singularity theory. No load-bearing gaps show up in the construction itself, but the result is incremental rather than foundational. Readers already tracking Monsky-type periodicity or positive-characteristic invariants will find the explicit families useful for examples. Others can safely skip it. The work is grounded enough and the evidence is reproducible enough that it should go to referees rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for every positive integer π there exists a one-dimensional local ring R_π in prime characteristic p such that the eventually periodic term φ in the Hilbert-Kunz function of R_π is immediately periodic with period π. This is achieved by an explicit, parameter-free construction of the rings followed by direct computation of the relevant lengths to verify that periodicity begins at the first possible index.

Significance. If the result holds, it strengthens Monsky's theorem by showing that every positive integer arises as an immediate period for φ, going beyond the known cases of period 1 (power series rings) and period 2 (examples of Kunz and Monsky). The explicit construction and direct verification without hidden divisibility conditions or unproven existence statements constitute a clear strength.

minor comments (2)

[Introduction] The introduction would benefit from a brief recall of the precise statement of Monsky's theorem on eventual periodicity before stating the main result.
Notation for the length function ℓ(R/I^n) could be standardized across the computations in the main construction section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately summarizes the main result and its relation to prior work of Monsky, Kunz, and Monsky.

Circularity Check

0 steps flagged

No significant circularity; explicit construction of rings

full rationale

The paper states an existence result extending Monsky's theorem on eventual periodicity of the Hilbert-Kunz function φ. It supplies an explicit, parameter-free construction of one-dimensional local rings R_π in prime characteristic such that φ is immediately periodic with any prescribed period π, verified by direct computation of lengths with no fitted parameters, no self-citations as load-bearing premises, and no reductions of predictions to inputs by definition. The derivation is self-contained against external benchmarks and does not rely on renaming, ansatz smuggling, or uniqueness theorems imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no visible free parameters, invented entities, or ad-hoc axioms beyond the background theorem of Monsky.

axioms (1)

domain assumption Monsky's theorem that the Hilbert-Kunz function of a one-dimensional local ring of prime characteristic has an eventually periodic term φ
The paper explicitly builds on this result to produce examples with prescribed immediate periods.

pith-pipeline@v0.9.0 · 5618 in / 1091 out tokens · 22262 ms · 2026-05-25T18:44:23.772546+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

[1]

P. G. L. Dirichlet, There are inﬁnitely many prime numbers in all arithmetic pro gressions with ﬁrst term and diﬀerence coprime , 1837, Translate by R. Stephan in 2014. Available on arXiv at https://arxiv.org/abs/0808.1408

work page internal anchor Pith review Pith/arXiv arXiv 2014
[2]

Kreuzer, Computing Hilbert–Kunz functions of 1-dimensional graded rings, Univ

M. Kreuzer, Computing Hilbert–Kunz functions of 1-dimensional graded rings, Univ. Iagel. Acta Math. 45 (2007), 81–95

work page 2007
[3]

Kunz, Characterizations of regular local rings of characteristi c p, Amer

E. Kunz, Characterizations of regular local rings of characteristi c p, Amer. J. Math. 91 (1969), 772–784

work page 1969
[4]

Monsky, The Hilbert–Kunz function , Math

P. Monsky, The Hilbert–Kunz function , Math. Ann. 263 (1983), 43–50

work page 1983
[5]

Upadhyay, The Hilbert–Kunz function for binomial hypersurfaces , Algebra 2014 (2014), Available at https://doi.org/10.1155/2014/525467

S. Upadhyay, The Hilbert–Kunz function for binomial hypersurfaces , Algebra 2014 (2014), Available at https://doi.org/10.1155/2014/525467. E-mail address : rbaidya@utk.edu Department of Mathematics, The University of Tennessee, Kn oxville, Tennessee 37996

work page doi:10.1155/2014/525467 2014

[1] [1]

P. G. L. Dirichlet, There are inﬁnitely many prime numbers in all arithmetic pro gressions with ﬁrst term and diﬀerence coprime , 1837, Translate by R. Stephan in 2014. Available on arXiv at https://arxiv.org/abs/0808.1408

work page internal anchor Pith review Pith/arXiv arXiv 2014

[2] [2]

Kreuzer, Computing Hilbert–Kunz functions of 1-dimensional graded rings, Univ

M. Kreuzer, Computing Hilbert–Kunz functions of 1-dimensional graded rings, Univ. Iagel. Acta Math. 45 (2007), 81–95

work page 2007

[3] [3]

Kunz, Characterizations of regular local rings of characteristi c p, Amer

E. Kunz, Characterizations of regular local rings of characteristi c p, Amer. J. Math. 91 (1969), 772–784

work page 1969

[4] [4]

Monsky, The Hilbert–Kunz function , Math

P. Monsky, The Hilbert–Kunz function , Math. Ann. 263 (1983), 43–50

work page 1983

[5] [5]

Upadhyay, The Hilbert–Kunz function for binomial hypersurfaces , Algebra 2014 (2014), Available at https://doi.org/10.1155/2014/525467

S. Upadhyay, The Hilbert–Kunz function for binomial hypersurfaces , Algebra 2014 (2014), Available at https://doi.org/10.1155/2014/525467. E-mail address : rbaidya@utk.edu Department of Mathematics, The University of Tennessee, Kn oxville, Tennessee 37996

work page doi:10.1155/2014/525467 2014