On the monomial reduction number of a monomial ideal in $K[x,y]$

Ali Soleyman Jahan; J\"urgen Herzog; Masoomeh Rahimbeigi; Somayeh Moradi

REVIEW 1 major objections 5 minor 10 references

On the monomial reduction number of a monomial ideal in $K[x,y]$

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Pith's one-line read This paper proves that a monomial ideal in $K[x,y]$ with monomial reduction $(x^a,y^b)$ has reduction number at most $g/\gcd(A)-|A|+2$, and classifies the ideals that attain the maximum $g-1$.

desk verdict A solid, correct classification of monomial reduction numbers in two variables; specialized but worth a real referee. read the letter →

arxiv 1908.03765 v1 pith:47TMYJKF submitted 2019-08-10 math.AC

Jürgen Herzog , Somayeh Moradi , Masoomeh Rahimbeigi , Ali Soleyman Jahan This is my paper · ORCID

classification math.AC MSC 13F2013H10

keywords monomialidealsreductionnumberquasi-equigeneratedfiberconeCastelnuovo-Mumfordregularitymultiplicitytwo-variablepolynomialring

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies monomial ideals in the two-variable polynomial ring $K[x,y]$ whose unique minimal monomial reduction is $J=(x^a,y^b)$, asking how many powers are needed before $I^{r+1}=JI^r$. For the quasi-equigenerated case, where every generator lies on the segment from $(a,0)$ to $(0,b)$, the ideal is encoded by a subset $A\subseteq\{0,\dots,g\}$ with $g=\gcd(a,b)$. The main result is the sharp bound $r(I)\le g/\gcd(A)-|A|+2$, proved by viewing the fiber cone as a projective curve and applying a regularity bound. The paper also shows every value $1\le j\le g-1$ occurs as a reduction number, classifies the ideals attaining the maximum $g-1$, and analyzes reduction numbers of powers of such ideals. If correct, it turns the computation of reduction numbers for this large class into a finite combinatorial problem in the exponent set $A$.

What carries the argument

The load-bearing object is the fiber cone $F(I)=\bigoplus_{k\ge0} I^k/\mathfrak{m}I^k$, a standard graded algebra whose maximal ideal has reduction number equal to $r(I)$ when $J=(x^a,y^b)$ is the monomial reduction. The key identity is $e(F(I_A))=g/\gcd(A)$, where $e$ denotes multiplicity; together with the Gruson--Lazarsfeld--Peskine theorem $\operatorname{reg}(F(I))\le e(F(I))-|A|+2$ and Trung's inequality $r\le\operatorname{reg}$, this yields the main bound. The subset $A\subseteq[0,g]$ encodes the generators, and sums of subsets correspond to products of ideals, so the argument reduces to counting lattice points on the segment between $(a,0)$ and $(0,b)$.

What would settle it

Compute $r(I_A)$ over $\mathbb{F}_2$ for $a=b=g$ and $A=\{0,1,g-1,g\}$; the theorem predicts $r(I_A)\le g-2$ and $r(I_A^k)=1$ for $k\ge g-2$, so if direct computation shows stabilization first happens later than $g-2$, the bound or its base-field reduction fails.

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Extended reading notes

Core claim

The central discovery is that for a quasi-equigenerated monomial ideal $I_A=(x^{ia/g}y^{b-ib/g}:i\in A)$, with $\{0,g\}\subseteq A\subseteq[0,g]$ and $g=\gcd(a,b)$, the monomial reduction number is governed by two elementary invariants of $A$: its greatest common divisor and its cardinality. Theorem 2.2 establishes $r(I_A)\le g/\gcd(A)-|A|+2$. The proof computes the multiplicity of the fiber cone as $e(F(I_A))=g/\gcd(A)$, applies the Gruson--Lazarsfeld--Peskine regularity bound to the associated projective curve, and uses Trung's inequality $r\le\operatorname{reg}$ after reducing to the complex numbers. Theorem 3.4 shows the bound is sharp: the only ideals with $r(I_A)=g-1$ are the three-generated ideals $(x^a,x^{ea/g}y^{b-eb/g},y^b)$ with $\gcd(e,g)=1$, together with $(x^a,y^b)$ when $g=1$. The paper further characterizes reduction number 1, computes reduction numbers for three-generated ideals, and proves stabilization results for powers $I^k$.

Load-bearing premise

The proof of the sharp bound assumes that the reduction number of a monomial ideal does not change when the base field is replaced, so that a regularity theorem proved over $\mathbb{C}$ can be applied over any field $K$.

Editorial extensions

If this is right

For any quasi-equigenerated ideal $I_A$ with $g=\gcd(a,b)$, the reduction number is at most $g/\gcd(A)-|A|+2$, and the weaker bound $r(I)<g/\gcd(A)$ follows directly from the multiplicity computation.
Every value $1\le j\le g-1$ occurs as a reduction number: Theorem 3.1 constructs $A=[0,1]\cup[j+1,g]$ with $r(I_A)=j$.
The maximum $g-1$ is attained exactly by the three-generated ideals $(x^a,x^{ea/g}y^{b-eb/g},y^b)$ with $\gcd(e,g)=1$, together with $(x^a,y^b)$ when $g=1$.
For equigenerated ideals with $a=b$, once $r(I^k)=1$ for some $k$, the reduction number remains 1 for all larger powers; for $k\ge a-2$ this happens exactly when $\{0,1,a-1,a\}\subseteq A$.
For three-generated ideals that are not quasi-equigenerated, the reduction number lies below $\min\{a,b\}$, depends only on the lattice point $p=(c,d)$, and as $b$ varies the attainable values fill $\{1,\dots,a-1\}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: because the main bound is stated purely in terms of $|A|$ and $\gcd(A)$, one could compute $r(I_A)$ exactly for all $A$ with a fixed $g$ by checking membership in the semigroup generated by $A$; such tables would test the paper's expectation that the proportion of ideals with reduction number $j$ tends to 0 as $a\to\infty$.
Editorial inference: the proof reduces to $\mathbb{C}$ using the unproved base-field independence of reduction numbers; if that independence can be proved directly for monomial ideals, the same bound would follow without the projective-curve regularity machinery and might generalize to more variables.
Editorial inference: the classification of maximal reduction number suggests a general principle that extremal reduction numbers force few generators; if this principle holds in higher dimensions, it would constrain which monomial ideals can have large reduction number in terms of their minimal generating sets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read. The paper is a thorough, correct study of the reduction number for monomial ideals in K[x,y] with respect to the diagonal reduction ideal J=(x^a,y^b). It's not a breakthrough, but it fills in the classification for the two-variable case and does so with explicit combinatorial arguments.

The genuinely new content: Theorem 2.2 gives the sharp bound r(I_A) ≤ g/gcd(A) - |A| + 2 for quasi-equigenerated ideals, with equality classified in Theorem 3.4. Theorem 3.2 classifies the r=1 case, Proposition 3.3 gives a clean formula for three-generated ideals, and Theorem 3.1 shows the whole range 1..g-1 is attained. Section 6 is the most surprising part: every I∈I_{a,b} has a unique smallest superideal L with r(L)=1, and for three-generated ideals you can write down its generators explicitly (Theorem 6.4). Section 7 on powers is a natural extension; the result that c(I)=inf{k: r(I^k)=1} can be any value from 1 to g-2 is a nice sharp statement.

The paper reads well. Proofs are largely self-contained and checkable. The reduction to C in Theorem 2.2 is legitimate: equality of monomial ideals is an equality of exponent sets, so the least r is field-independent. The use of Gruson-Lazarsfeld-Peskine and Trung is standard and cited correctly. I found no load-bearing error, just minor typos. The citation pattern is honest; the only self-citation (Herzog–Hibi in Cor 7.3) is an independent published bound.

Soft spots: the scope is deliberately narrow. The asymptotic questions in Section 3 are left mostly open—for general j the limit is only conjectured to be 0, and Prop 3.5 shows the three-generated analogue oscillates. The question whether r(I^{k+1}) ≤ r(I^k) is acknowledged as an open problem. These are honest limitations, not flaws.

Who it's for: specialists in monomial ideals, Rees algebras, and fiber cones. It deserves a serious referee; I'd send it out. Expect acceptance after minor revision.

Referee Report

1 major / 5 minor

Summary. The paper investigates the monomial reduction number r(I) of monomial ideals I in K[x,y] for which J=(x^a,y^b) is the minimal monomial reduction. The authors introduce the class I_{a,b} and its quasi-equigenerated subclass parameterized by subsets A of {0,...,g}, where g=gcd(a,b). The central results are: (1) the bound r(I) ≤ g/gcd(A) - |A| + 2 for quasi-equigenerated I=I_A (Theorem 2.2), obtained from the Gruson-Lazarsfeld-Peskine regularity bound and Trung's reduction-number bound; (2) Theorem 3.1 showing every value 1,...,g-1 occurs; (3) Theorem 3.2 characterizing r=1 and Theorem 3.4 classifying the ideals attaining the maximal value g-1; (4) results for three-generated ideals including formulas for r(I_p) and the set R(a,b); (5) the construction in Theorem 6.4 of the unique smallest overideal of reduction number 1; and (6) results on reduction numbers of powers, including Theorem 7.2 and Proposition 7.5. The proofs are mostly self-contained combinatorial arguments, with standard external tools correctly cited.

Significance. If correct, the paper gives a sharp, generator-dependent upper bound for the reduction number of quasi-equigenerated monomial ideals in two variables and a complete classification of the extremal cases. The computations of the fiber cone multiplicity e(F(I_A))=g/gcd(A) and the use of the GLP regularity bound are clean and convincing. The paper is also useful for its explicit examples and for the analysis of reduction numbers of powers. I note that the base-field step in Theorem 2.2 ('We may assume K=C') is valid: the equality I^{r+1}=JI^r is an equality of monomial ideals, hence depends only on exponent sets. However, one peripheral statement, Proposition 7.5, is false as stated; this does not affect the paper's main classifications but requires correction.

major comments (1)

[Section 7, Proposition 7.5] Proposition 7.5 is false as stated. Let a=4 and I=(x^4,y^4,x^3y^2). Then I belongs to I_{4,4}; its equigenerated part is I_A=(x^4,y^4), and neither xy^3 nor x^3y is in I, so the 'otherwise' branch applies and the proposition predicts r(I^k)=2 for all k≥2. However, I^2=JI with J=(x^4,y^4), so r(I)=1. By Proposition 7.1, r(I^2)≤ceil((1-1)/2)+1=1; since I^2≠(x^8,y^8) (for instance, x^7y^2∈I^2 is not in (x^8,y^8)), we have r(I^2)=1, contradicting the predicted value 2. The proof's lower bound r(I^k)≥r(I_A^k)≥2 fails when I_A=(x^a,y^a) or, more generally, when gcd(A)>1, because Theorem 7.2 assumes gcd(A)=1.

minor comments (5)

[Section 7, Corollary 7.3] The last sentence of the proof, 'Hence the desired conclusion follow from Theorem 1.6', should cite the Trung bound used in Theorem 2.2 rather than Theorem 1.6, since Theorem 1.6 bounds r(I) by the arithmetic degree, not by the regularity.
[Section 4, equation (3)] Equation (3) has a missing parenthesis and misparenthesization; it should read r(I)=g/gcd(cg/a,g)-1 when ad+bc=ab.
[Section 3, Proposition 3.5] In the proof, 'na(d−1)=φ(a)' should be 'na(a−1)=φ(a)', since the ideal count concerns reduction number a−1.
[Section 7, Proposition 7.4] In the proof of (ii), the line containing '(i−1−i)d' is garbled and should read '(j−1−i)g' in the upper endpoint of the interval.
[Section 4, Proposition 4.5] The proof says 'there exist integers j,k>0', but the constructed choices allow j=0 or k=0; the statement should say 'nonnegative integers'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all load-bearing arguments derive from definitions, external theorems (Vasconcelos, GLP, Trung, Singla, Hoa), and independent published results; the only self-citation is auxiliary and not load-bearing.

full rationale

The central results are genuine derivations. Theorem 2.1 combines Vasconcelos's bound r(I) < arith-deg(F(I)) with an independent multiplicity computation; Theorem 2.2 applies the external Gruson–Lazarsfeld–Peskine regularity bound and Trung's result, after observing that the reduction number of a monomial ideal is base-field independent, since I^{r+1}=JI^r is an equality of monomial ideals and the least such r depends only on exponent sets. The combinatorial criteria in Theorems 3.1, 3.2, 3.4 and Proposition 3.3 are proven from the definition I^{r+1}=JI^r by explicit sumset computations, not assumed. Section 4 derives r(I_p)=min{k: u_p^k in J^k}-1 from the definition and uses it to compute values; no fitted parameter is renamed as a prediction. The one self-citation is [3, Theorem 1.1] in Corollary 7.3, used only to improve the bound by 1 under extra hypotheses; it is an externally published theorem with stated assumptions that do not include the target inequality, and the same conclusion is also attributed to the independent GLP bound. Therefore none of the central claims reduces to its own inputs, and no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper relies on a number of standard theorems from commutative algebra and number theory, listed above. It introduces no new objects or entities; its contribution is new theorems about existing invariants. No free parameters are fitted to data.

assumptions (8)

standard math Singla's theorem: the unique minimal monomial reduction ideal of a monomial ideal I is generated by the monomials corresponding to the 0-dimensional faces of conv(I).
Used in Section 1 (Proposition 1.2) to justify that J=(x^a,y^b) is the unique minimal monomial reduction ideal when all generators lie in the half-space H^+_{a,b}.
standard math Vasconcelos' bound: the reduction number of the graded maximal ideal of a standard graded algebra is less than its arithmetic degree.
The basis for Theorem 1.6 and hence the bound r(I) < arith-deg(F(I)).
standard math Associativity formula for multiplicities: e(A) = arith-deg(A) when all associated primes have maximal dimension.
Used in Remark 1.7 to replace arithmetic degree by multiplicity for the domain F(I).
standard math Gruson-Lazarsfeld-Peskine theorem: a regularity bound for irreducible curves in projective space.
Applied in Theorem 2.2 to bound reg(F(I)) by e(F(I)) - |A| + 2.
standard math Trung's inequality: r_{bar J}(m_{F(I)}) ≤ reg F(I).
Used in Theorem 2.2 to convert the regularity bound into a reduction number bound.
standard math Hoa's inequality: r(I^k) ≤ ceil((r(I)-1)/k)+1.
Used in Proposition 7.1 and in the proof of Proposition 7.5.
standard math Hardy-Wright theorems: lim sup_{a→∞} φ(a)/a = 1 and lim inf_{a→∞} (φ(a) log log a)/a = e^{-γ}.
Used in Proposition 3.5 to show that the limit of n_a(a-1)/n_a does not exist.
standard math Herzog-Hibi theorem: Castelnuovo-Mumford regularity bound for simplicial semigroup rings.
Used in Corollary 7.3 to improve the bound by 1 under additional assumptions.

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Pith. "Pith review of On the monomial reduction number of a monomial ideal in $K[x,y]$." pith.science (2026). https://pith.science/paper/47TMYJKF

@misc{pith2026190803765,
  author       = {Pith},
  title        = {Pith review of: On the monomial reduction number of a monomial ideal in $K[x,y]$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/47TMYJKF}},
  note         = {Machine review of arXiv:1908.03765}
}

abstract

The reduction number of monomial ideals in the polynomial $K[x,y]$ is studied. We focus on ideals $I$ for which $J=(x^a,y^b)$ is a reduction ideal. The computation of the reduction number amounts to solve linear inequalities. In some special cases the reduction number can be explicitly computed.

Figures

Figures reproduced from arXiv: 1908.03765 by the authors.

**Figure 1.** conv(I) for I = (x 7 , x6 y 2 , x3 y 3 , x2 y 5 , xy6 , y10). For any monomial u ∈ K[x, y], u = x c y d , let ν(u) = (bc + ad)/ab. Note that for any two monomials u, v ∈ K[x, y] one has ν(uv) = ν(u) + ν(v). We denote by Ia,b the set of monomial ideals I ⊂ K[x, y] with x a , yb ∈ G(I) and ν(u) ≥ 1 for all u ∈ G(I). By Corollary 1.3, the monomial ideals I ∈ Ia,b are precisely the monomial ideals in K[x, y] for which J… view at source ↗

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Bruns and J

W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in advanced mathematics 39, Cambridge University Press, Cambridge, UK, 1998

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[2]

Gruson, R

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[3]

Herzog and T

J. Herzog and T. Hibi, Castelnuovo-Mumford regularity of simplicia l semigroup rings with isolated singularity, Proc. Amer. Math. Soc. 131, (2003) 2641–2647

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Hoa, Reduction numbers and Rees algebras of powers of an id eal, Proc

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Hoa and J

L.T. Hoa and J. St¨ uckrad, Castelnuovo-Mumford regularity ofsimplicial toric rings, J. Algebra 259, (2003) 127–146

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G. H Hardy and E. M. Wright, An Introduction to the Theory of Nu mbers (Fifth ed.), Oxford University Press, (1979)

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I. Swanson and C. Huneke, Integral closure of ideals, rings, and modules , London Mathematical Society Lecture Note Series 336. Cambridge University Press, 2006

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Vasconcelos, The reduction number of an algebra, Composit io Mathematica, 104, (1996) 189–197

W. Vasconcelos, The reduction number of an algebra, Composit io Mathematica, 104, (1996) 189–197. J¨urgen Herzog, F achbereich Mathematik, Universit ¨at Duisburg-Essen, Campus Essen, 45117 Essen, Germany E-mail address : juergen.herzog@uni-essen.de Somayeh Moradi, Department o...

1996

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bruns and J

W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in advanced mathematics 39, Cambridge University Press, Cambridge, UK, 1998

work page 1998

[2] [2]

Gruson, R

L. Gruson, R. Lazarsfeld and C. Peskine, On a theorem of Caste lnuovo, and the equations deﬁning space curves, Invent. Math. 72, (1983) 491–506

work page 1983

[3] [3]

Herzog and T

J. Herzog and T. Hibi, Castelnuovo-Mumford regularity of simplicia l semigroup rings with isolated singularity, Proc. Amer. Math. Soc. 131, (2003) 2641–2647

work page 2003

[4] [4]

Hoa, Reduction numbers and Rees algebras of powers of an id eal, Proc

L.T. Hoa, Reduction numbers and Rees algebras of powers of an id eal, Proc. Amer. Math. Soc. 119, (1993) 415–422

work page 1993

[5] [5]

Hoa and J

L.T. Hoa and J. St¨ uckrad, Castelnuovo-Mumford regularity ofsimplicial toric rings, J. Algebra 259, (2003) 127–146

work page 2003

[6] [6]

H Hardy and E

G. H Hardy and E. M. Wright, An Introduction to the Theory of Nu mbers (Fifth ed.), Oxford University Press, (1979)

work page 1979

[7] [7]

Singla, Minimal monomial reductions and the reduced ﬁber ring o f an extremal ideal, Illinois J

P. Singla, Minimal monomial reductions and the reduced ﬁber ring o f an extremal ideal, Illinois J. Math. 51, (2007) 1085–1102

work page 2007

[8] [8]

Trung, Reduction exponent and degree bound for the deﬁn ing equations of graded rings, Proc

N.V. Trung, Reduction exponent and degree bound for the deﬁn ing equations of graded rings, Proc. Amer. Math. Soc. 101, (1987) 229–236

work page 1987