New bounds on Castelnuovo--Mumford regularity of monomial curves and application to sumsets

Doan Quang Tien, Le Tuan Hoa

Pith reviewed 2026-05-09 23:27 UTC · model grok-4.3

classification 🧮 math.AC

keywords monomial curvesCastelnuovo-Mumford regularityApery setsFrobenius numbersCohen-Macaulay propertyBuchsbaum ringsnumerical semigroupssumsets

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The pith

Under specific conditions, Apery-set and Frobenius-number arguments yield bounds on monomial curve regularity stricter than the sum of the two largest gaps plus two.

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

The paper derives new upper bounds on the Castelnuovo-Mumford regularity of monomial curves defined by sequences of coprime integers that satisfy certain conditions. These bounds improve upon Lvovsky's bound by using the structure of the Apery set and the Frobenius number rather than simply summing gaps. The method also supplies criteria for when the curve ring is arithmetically Cohen-Macaulay or Buchsbaum and includes algorithms for verification and computation. An application shows how the regularity controls the additive structure of sumsets generated by the sequence.

Core claim

When the sequence satisfies the paper's conditions, the Castelnuovo-Mumford regularity of the monomial curve is bounded by a quantity derived from its Apery set that is smaller than the sum of its two largest gaps plus two. This follows from relating the highest degree of a minimal generator of the syzygy module to the maximal non-element in the semigroup or properties of the Apery set, extending the gap-based estimates of Gruson-Lazarsfeld-Peskine and Lvovsky.

What carries the argument

Apery sets and Frobenius numbers of the numerical semigroup generated by the coprime sequence, used to bound the Castelnuovo-Mumford regularity via the degrees appearing in the minimal free resolution.

Load-bearing premise

The sequence of coprime integers must satisfy the specific conditions under which the Apery-set and Frobenius-number analysis produces a bound tighter than the sum of the two largest gaps.

What would settle it

Find a sequence of coprime integers that meets the paper's conditions but whose monomial curve has Castelnuovo-Mumford regularity larger than the new bound derived from its Apery set.

read the original abstract

A monomial curve $C$ is defined by a sequence of coprime integers $0 = a_0 < a_1 < \cdots < a_k =: d$. One gap of this sequence is $a_{i+1} - a_i - 1$. Gruson--Lazarsfeld--Peskine bound (1983) says that $reg (C) \le d - k +2$, which is equal to the sum of all gaps plus 2. Lvovsky (1996) showed that it is enough to take the sum of two largest gaps plus 2. In this paper, under some specific conditions, we give several new bounds which are better than Lvovsky's bound. Our method relies on the study of Apery sets and Frobenius numbers. From this we can give new criteria to check the (arithmetically) Cohen--Macaulay and Buchsbaum property of $C$. Algorithms are provided to check these properties as well as to compute $ reg(C)$ and other invariants. We also give an application to study the structure of sumsets.

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

Improves Lvovsky's regularity bound for monomial curves under explicit conditions via Apery sets, with added algorithms and a sumset corollary.

read the letter

The main thing to know is that this paper improves Lvovsky's bound on the Castelnuovo-Mumford regularity for monomial curves under specific conditions using Apery sets and Frobenius numbers. It also gives algorithms for checking Cohen-Macaulay and Buchsbaum properties and computing the regularity, plus an application to sumsets. The derivations hold up. They use standard semigroup facts to get stricter upper bounds when the conditions are met, and the pseudocode follows directly without loose ends. The sumset part is a clean corollary. The conditions restrict the scope, so the better bounds do not apply universally. The paper could benefit from more examples showing the size of the improvement in practice, but that is a minor point rather than a problem with the logic. This work is for people in commutative algebra focused on monomial curves and numerical semigroups. A reader looking for computational tools or refined bounds in this niche will get something usable. The math is grounded enough to warrant a serious referee. I would send it out for peer review.

Referee Report

0 major / 2 minor

Summary. The paper claims that for monomial curves defined by coprime integer sequences 0 = a0 < a1 < ⋯ < ak = d, new upper bounds on Castelnuovo-Mumford regularity reg(C) can be obtained that are strictly smaller than Lvovsky's sum-of-two-largest-gaps + 2, under explicitly stated conditions on the sequence. These bounds are derived from the cardinality and structure of the Apery set and the Frobenius number of the associated numerical semigroup. The work also supplies pseudocode algorithms for detecting arithmetically Cohen-Macaulay and Buchsbaum properties of C, for computing reg(C), and applies the regularity results to describe the structure of sumsets.

Significance. If the derivations hold, the results tighten the known bounds on regularity for a nontrivial subclass of monomial curves and supply practical algorithms that follow directly from standard numerical-semigroup theory. The explicit conditions (primarily in §§3–4) and the direct corollary for sumsets are strengths; the paper thereby offers both a theoretical improvement over Gruson–Lazarsfeld–Peskine and Lvovsky and concrete computational tools.

minor comments (2)

[§5] §5: the pseudocode for the regularity-computation algorithm would benefit from a short remark linking each step to the Apery-set cardinality formula used in the preceding theorem.
[Theorem 4.3] The statement of the main bounds in Theorem 4.3 could include a one-sentence reminder of the precise hypothesis on the gaps that distinguishes it from Lvovsky's result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; derivations use independent semigroup theory

full rationale

The paper derives improved bounds on Castelnuovo-Mumford regularity from explicit Apery-set cardinality and Frobenius-number formulas applied to numerical semigroups generated by the given coprime sequence. These are standard external tools whose definitions and properties predate the paper and do not reference the target regularity bounds. Sections 3 and 4 state the precise conditions on the sequence, carry out the derivations to obtain strict inequalities versus Lvovsky's sum-of-two-largest-gaps bound, and supply pseudocode algorithms that follow directly from the same semigroup invariants. The sumset application is a straightforward corollary. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard facts about numerical semigroups generated by coprime integers, the definition and basic properties of Apery sets, and the Frobenius number; no free parameters are introduced and no new entities are postulated.

axioms (2)

domain assumption The sequence 0 = a0 < a1 < ... < ak = d consists of coprime positive integers and therefore generates a numerical semigroup.
Invoked in the definition of the monomial curve and throughout the study of gaps, Apery sets, and Frobenius numbers.
standard math Standard properties of Apery sets and the Frobenius number hold for numerical semigroups.
Used as the basis for deriving the new regularity bounds.

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Works this paper leans on

25 extracted references · 17 canonical work pages

[1]

Bermejo and P

I. Bermejo and P. Gimenez,Saturation and Castelnuovo–Mumford regularity. J. Algebra 303(2006), 592–617. doi.org/10.1016/j.jalgebra.2005.05.020

work page doi:10.1016/j.jalgebra.2005.05.020 2006
[2]

Brauer,On a problem of partitions

A. Brauer,On a problem of partitions. Amer. J. Math.64(1942), 299–312. doi.org/10.2307/2371684

work page doi:10.2307/2371684 1942
[3]

Bresinsky,Monomial Buchsbaum ideals inP r

H. Bresinsky,Monomial Buchsbaum ideals inP r. Manuscripta Math.47(1984), 105–132

1984
[4]

Bresinsky, F

H. Bresinsky, F. Curtis, M. Fiorentini and L. T. Hoa,On the structure of local cohomology modules for monomial curves inP 3 K. Nagoya Math. J.136(1994), 81–114

1994
[5]

Elias,Sumsets and Projective Curves

J. Elias,Sumsets and Projective Curves. Mediterr. J. Math.19(2022), no. 4, Paper No. 177, 11 pp. doi.org/10.1007/s00009-022-02108-0. 23

work page doi:10.1007/s00009-022-02108-0 2022
[6]

Erd¨ os and R

P. Erd¨ os and R. L. Graham,On a linear diophantine problem of Frobenius. Acta Arith.21 (1972), 399–408

1972
[7]

Fiorentini and L

M. Fiorentini and L. T. Hoa,On monomial k-Buchsbaum curves inP r. Ann. Univ. Ferrara Sez. VII (N.S.)36(1990), 159–174 (1991)

1990
[8]

Gimenez and M

P. Gimenez and M. Gonz´ alez -S´ anchez,Castelnuovo-Mumford regularity of projective mono- mial curves via sumsets. Mediterr. J. Math.20(2023), no. 5, Paper No. 287, 24 pp. doi.org/10.1007/s00009-023-02482-3

work page doi:10.1007/s00009-023-02482-3 2023
[9]

S. Goto, N. Suzuki and K. Watanabe,On affine semigroup rings. Japan. J. Math. (N.S.)2 (1976), 1–12. doi.org/10.4099/math1924.2.1

work page doi:10.4099/math1924.2.1 1976
[10]

Granville and A

A. Granville and A. Walker,A tight structure theorem for sumsets. Proc. Amer. Math. Soc. 149(2021), 4073–4082

2021
[11]

Gruson, R

L. Gruson, R. Lazarsfeld and C. Peskine,On a theorem of Castelnuovo, and the equations defining space curves. Invent. Math.72(1983), 491–506. doi.org/10.1007/BF01398398

work page doi:10.1007/bf01398398 1983
[12]

Hellus, L

M. Hellus, L. T. Hoa and J. St¨ uckrad,Castelnuovo-Mumford regularity and reduction number of some monomial curves. Proc. Amer. Math. Soc.138(2010), 27–35.http: //www.jstor.org/stable/40590591

work page arXiv 2010
[13]

Herzog,Generators and relations of abelian semigroups and semigroup rings

J. Herzog,Generators and relations of abelian semigroups and semigroup rings. Manuscripta Math.3(1970), 175–193. doi.org/10.1007/BF01273309

work page doi:10.1007/bf01273309 1970
[14]

Herzog and D

J. Herzog and D. I. Stamate,Cohen-Macaulay criteria for projective monomial curves via Gr¨ obner bases. Acta Math. Vietnam.44(2019), 51–64. doi.org/10.1007/s40306-018-00302- 5

work page doi:10.1007/s40306-018-00302- 2019
[15]

L. T. Hoa and C. Miyazaki,Bounds on Castelnuovo-Mumford regularity for generalized Cohen-Macaulay graded rings. Math. Ann.301(1995), 587–598. doi.org/10.1007/BF01446647

work page doi:10.1007/bf01446647 1995
[16]

K¨ astner,Zu einem Problem von H

J. K¨ astner,Zu einem Problem von H. Bresinsky ¨ uber monomiale Buchsbaum Kurven. Manuscripta Math.54(1985), 197–204. doi.org/10.1007/BF01171707

work page doi:10.1007/bf01171707 1985
[17]

T. T. G. Lam and N. V. Trung,Buchsbaumness and Castelnuovo-Mumford regularity of non-smooth monomial curves. J. Algebra590(2022), 313–337. doi.org/10.1016/j.jalgebra.2021.10.008

work page doi:10.1016/j.jalgebra.2021.10.008 2022
[18]

Lvovsky,On inflection points, monomial curves, and hypersurfaces containing projective curves

S. Lvovsky,On inflection points, monomial curves, and hypersurfaces containing projective curves. Math. Ann.306(1996), 719–735. doi.org/10.1007/BF01445273

work page doi:10.1007/bf01445273 1996
[19]

M. B. Nathanson, Additive Number Theory. Inverse Problems and the Geometry of Sum- sets. Graduate Texts in Mathematics, vol. 165. Springer, New York (1996)

1996
[20]

M. J. Nitsche,A combinatorial proof of the Eisenbud-Goto conjecture for mono- mial curves and some simplicial semigroup rings. J. Algebra397(2014), 47–67. doi.org/10.1016/j.jalgebra.2013.08.026. 24

work page doi:10.1016/j.jalgebra.2013.08.026 2014
[21]

J. L. Ram´ ırez Alfonsin, The Diophantine Frobenius problem. Oxford Lecture Series in Mathematics and its Applications, 30. Oxford University Press, Oxford, 2005. xvi+243 pp

2005
[22]

E. S. Selmer,On the linear Diophantine problem of Frobenius. J. Reine Angew. Math. 293(294)(1977), 1–17

1977
[23]

Stanley,Hilbert functions of graded algebras

R. Stanley,Hilbert functions of graded algebras. Advances in Math.28(1978), 57–83. doi.org/10.1016/0001-8708(78)90045-2

work page doi:10.1016/0001-8708(78)90045-2 1978
[24]

N. V. Trung,Projections of one-dimensional Veronese varieties. Math. Nachr.118(1984), 47–67. doi.org/10.1002/mana.19841180104

work page doi:10.1002/mana.19841180104 1984
[25]

N. V. Trung and L. T. Hoa,Affine semigroups and Cohen-Macaulay rings generated by monomials. Trans. Amer. Math. Soc.298(1986), 145–167. doi.org/10.2307/2000613. APPENDIX In this section we construct several algorithms. In the first algorithm, given a subsetAwe compute the Ap´ ery setAp(A) as a vector inN d and degrees of its elements as a vector Deg(A)∈N ...

work page doi:10.2307/2000613 1986