pith. sign in

arxiv: 2507.17935 · v3 · submitted 2025-07-23 · 🧮 math.AC

On the Stanley length of monomial ideals

Mircea Cimpoeas This is my paper

Pith reviewed 2026-05-19 03:10 UTC · model grok-4.3

classification 🧮 math.AC
keywords monomial idealsStanley decompositionStanley lengthlinear quotientspolynomial ringscommutative algebra
0
0 comments X

The pith

Monomial ideals have a Stanley length bounded above by a function of their minimal generators.

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

The paper studies the Stanley length of a monomial ideal, which is the smallest number of summands needed in a Stanley decomposition of the ideal as a multigraded module over the polynomial ring. It proves a general upper bound on this length expressed directly in terms of the minimal monomial generators. Exact formulas are derived when the ring has two variables or the ideal has exactly two minimal generators. The length equals the number of minimal generators precisely when the ideal has linear quotients, with the converse holding in some special cases.

Core claim

Let I be a monomial ideal in K[x1,...,xn] minimally generated by m monomials. Then slength(I) admits an upper bound in terms of these generators. When n=2 or m=2, slength(I) has an explicit formula. If I has linear quotients then slength(I)=m, and the converse holds in some special cases.

What carries the argument

Stanley length (slength), the minimal number of summands appearing in any Stanley decomposition of the ideal viewed as a multigraded module.

If this is right

  • Monomial ideals with linear quotients achieve Stanley length exactly equal to their number of minimal generators.
  • The Stanley length of any monomial ideal in two variables admits an explicit formula.
  • The Stanley length of any monomial ideal with exactly two minimal generators admits an explicit formula.
  • An explicit upper bound on Stanley length holds for every monomial ideal in any number of variables.

Where Pith is reading between the lines

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

  • The bound may simplify estimates of homological invariants such as depth or Castelnuovo-Mumford regularity for the quotient by the ideal.
  • The connection to linear quotients suggests that similar length results could be tested for other classes of ideals that admit simple resolutions.
  • The formulas in the two-variable and two-generator cases provide test cases for possible generalizations to three or more generators.

Load-bearing premise

Every monomial ideal admits at least one Stanley decomposition of finite length, so the minimal length is well-defined.

What would settle it

A concrete monomial ideal whose Stanley length exceeds the upper bound stated in terms of its minimal generators.

read the original abstract

Let $S=K[x_1,\ldots,x_n]$ be the ring of polynomials in $n$ variables over an arbitrary field $K$. Given a finitely generated multigraded module $M$, its Stanley length, denoted by $\operatorname{slength}(M)$, is the minimal length of a Stanley decomposition of $M$. Let $I\subset S$ be a monomial ideal, minimally generated by $m$ monomials. We give an upper bound for $\operatorname{slength}(I)$, in terms of its minimal monomial generators. Also, we give precise formulas for $\operatorname{slength}(I)$, if $n=2$ or $m=2$. Also, we show that if $I$ has linear quotients, then $\operatorname{slength}(I)=m$, and the converse holds in some special cases.

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

0 major / 2 minor

Summary. The paper studies the Stanley length slength(I) of a monomial ideal I in S = K[x_1,…,x_n], defined as the minimal number of summands in a Stanley decomposition of I as a multigraded S-module. For I minimally generated by m monomials, the authors prove an upper bound on slength(I) expressed in terms of the minimal generators. They derive explicit formulas for slength(I) when n=2 or m=2. They also show that slength(I)=m whenever I has linear quotients, with the converse holding under additional restrictions on the ideal or the ring.

Significance. The results supply concrete, computable bounds and equalities for the Stanley length of monomial ideals, linking this invariant to the well-studied property of linear quotients. Such formulas can aid explicit calculations in combinatorial commutative algebra and may serve as test cases for broader conjectures on Stanley decompositions.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'the converse holds in some special cases' is vague; a parenthetical indication of the precise hypotheses (e.g., 'when the generators form a regular sequence' or 'when n=2') would improve readability without lengthening the abstract.
  2. [Introduction] Notation: the manuscript uses both 'minimal monomial generators' and 'minimal generators' interchangeably; a single consistent phrase or a short glossary entry would prevent minor confusion for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation of minor revision. The summary accurately reflects the main results of the paper: an upper bound on slength(I) for a monomial ideal with m minimal generators, explicit formulas when n=2 or m=2, and the equality slength(I)=m when I has linear quotients (with converse in special cases).

Circularity Check

0 steps flagged

No significant circularity; results follow from standard definitions and explicit constructions

full rationale

The paper recalls the standard definition of Stanley length as the minimal number of summands in a Stanley decomposition of a finitely generated multigraded module over a polynomial ring, which exists and is finite by the Noetherian property. It then derives an upper bound on slength(I) in terms of the m minimal monomial generators, exact formulas when n=2 or m=2, and the relation slength(I)=m when I has linear quotients (with partial converse). These follow from direct combinatorial arguments on monomial generators and inductive constructions that do not reduce any claim to a fitted input, self-definition, or load-bearing self-citation. The central results remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The work rests on standard definitions from commutative algebra and the existence of minimal monomial generating sets; no free parameters or new entities are introduced.

axioms (3)
  • standard math S = K[x1, …, xn] is the polynomial ring over an arbitrary field K
    Opening sentence of the abstract; standard setup for monomial ideal theory.
  • domain assumption I is a monomial ideal minimally generated by m monomials
    Central object of study; required for all stated bounds and formulas.
  • domain assumption Every finitely generated multigraded module admits a Stanley decomposition whose length is finite
    Implicit in the definition of slength(M) given in the abstract.

pith-pipeline@v0.9.0 · 5664 in / 1419 out tokens · 53986 ms · 2026-05-19T03:10:47.022468+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

12 extracted references · 12 canonical work pages

  1. [1]

    Anwar,Janet’s Algorithm, Bull

    I. Anwar,Janet’s Algorithm, Bull. Math. Soc. Sci. Math. Roumanie51(99)(1)(2008), 11–19

    work page 2008

  2. [2]

    Apel,On a conjecture of R

    J. Apel,On a conjecture of R. P. Stanley; Part II - Quotients Modulo Monomial Ideals, J. of Alg. Comb.17(2003), 57–74

    work page 2003

  3. [3]

    Cimpoea¸ s,Several inequalities regarding Stanley depth, Romanian Journal of Mathematics and Computer Science2(1), (2012), 28–40

    M. Cimpoea¸ s,Several inequalities regarding Stanley depth, Romanian Journal of Mathematics and Computer Science2(1), (2012), 28–40

    work page 2012

  4. [4]

    A. M. Duval, B. Goeckneker, C. J. Klivans, J. L. Martine,A non-partitionable Cohen-Macaulay sim- plicial complex, Advances in Mathematics, vol299(2016), pag. 381–395

    work page 2016

  5. [5]

    S. A. Seyed Fakhari,Stanley depth and symbolic powers of monomial ideals, Math. Scand.120(2017), 5–16

    work page 2017

  6. [6]

    Ichim, L

    B. Ichim, L. Katth¨ an, J.J. Moyano-Fernand´ ez,The behavior of Stanley depth under polarization, Journal of Combinatorial Theory, Series A135(2015), 332–347

    work page 2015

  7. [7]

    Ishaq,Upper bounds for Stanley depth, Comm

    M. Ishaq,Upper bounds for Stanley depth, Comm. Algebra40(1)(2012), 87–97

    work page 2012

  8. [8]

    Herzog, T

    J. Herzog, T. Hibi,Monomial Ideals, Graduate Texts in Mathematics, vol. 260, Springer- Verlag London, Ltd., London, 2011

    work page 2011

  9. [9]

    Herzog, M

    J. Herzog, M. Vladoiu, X. Zheng,How to compute the Stanley depth of a monomial ideal, Journal of Algebra322(9), (2009), 3151–3169

    work page 2009

  10. [10]

    Popescu,Bounds of Stanley depth, An

    D. Popescu,Bounds of Stanley depth, An. S ¸t. Univ. Ovidius Constant ¸a19(2)(2011), 187–194

    work page 2011

  11. [11]

    Rauf,Depth and sdepth of multigraded modules, Communications in Algebra,vol

    A. Rauf,Depth and sdepth of multigraded modules, Communications in Algebra,vol. 38, Issue 2, (2010), 773–784

    work page 2010

  12. [12]

    R. P. Stanley,Linear Diophantine equations and local cohomology, Invent. Math.68(1982), pag. 175–193

    work page 1982