Prescribed Initial Behavior of μ(I^k)
Pith reviewed 2026-06-26 06:20 UTC · model grok-4.3
The pith
Any prescribed finite pattern of increases, decreases, or equalities can occur in the first differences of μ(I^k) for monomial ideals in two variables.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For any finite sequence of signs (positive, negative, or zero), there exists a monomial ideal I in K[x,y] such that the differences μ(I^{k+1}) - μ(I^k) match that sign pattern for the initial terms, before becoming positive.
What carries the argument
A monomial ideal in two variables whose minimal generators are selected to control the differences in the number of minimal generators of its successive powers independently for each step.
If this is right
- The initial behavior of μ(I^k) is completely flexible in two variables.
- Sign-realization holds for the index of reducibility of powers of m-primary ideals in two-dimensional regular local rings.
- Known examples of non-monotonic μ(I^k) are special cases of this general construction.
Where Pith is reading between the lines
- This suggests that similar flexibility might be possible in higher dimensions or for other functions like Betti numbers.
- Algebraic computations involving powers may need to account for arbitrary initial fluctuations rather than assuming monotonicity from the start.
- Connections could exist to combinatorial objects like numerical semigroups or lattice points in two dimensions.
Load-bearing premise
That a monomial ideal in two variables exists whose minimal generators can be chosen to make each initial difference μ(I^{k+1})-μ(I^k) have any desired sign independently.
What would settle it
A specific finite sign sequence for which no monomial ideal in two variables produces the corresponding pattern in the first differences of μ(I^k).
read the original abstract
It is well known that for every graded ideal $I$, the numbers of minimal generators $\mu(I^k)$ of its powers of $I$ are eventually increasing. However, its initial behavior can be surprisingly flexible. We prove that any prescribed finite pattern of increases, decreases, and equalities can occur among the first differences $\mu(I^{k+1})-\mu(I^k)$ of a suitable monomial ideal $I$ in $K[x,y]$. This provides a broad positive answer to a previously posed sign-realization problem and unifies several known constructions exhibiting unusual behavior of $\mu(I^k)$. Moreover, as a consequence, analogous sign-realization results are obtained for the index of reducibility of powers of $\mathfrak m$-primary ideals in two-dimensional regular local rings.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that any finite prescribed pattern of increases (+), decreases (-), and equalities (=) can appear in the initial segment of the sequence of first differences δ_k = μ(I^{k+1}) − μ(I^k) for a monomial ideal I ⊂ K[x,y], before the known eventual positivity and monotonicity of the differences takes over. The argument proceeds by explicit construction of the minimal generators of I whose exponent vectors are chosen to control the appearance of new minimal generators in successive powers I^k. As a corollary, the same sign patterns are realized for the index of reducibility of powers of m-primary ideals in two-dimensional regular local rings.
Significance. The result supplies a complete affirmative answer to the sign-realization question for the initial behavior of μ(I^k), unifying earlier ad-hoc examples that exhibited decreases or plateaus. The explicit monomial construction in exactly two variables is a notable strength, as it shows that the eventual increase does not constrain the finite initial segment. The extension to the index of reducibility broadens the applicability within commutative algebra.
minor comments (2)
- [Abstract] Abstract, line 3: the phrase 'previously posed sign-realization problem' lacks a citation; adding the reference would clarify the context of the contribution.
- [Introduction] The notation μ(I^k) is used throughout without an explicit reminder that it denotes the minimal number of generators; a parenthetical definition on first use would aid readability.
Simulated Author's Rebuttal
We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.
Circularity Check
No significant circularity identified
full rationale
The paper is an existence proof establishing that arbitrary finite sign patterns (increases, decreases, equalities) can appear in the initial segment of the sequence μ(I^{k+1}) − μ(I^k) for monomial ideals I in K[x,y]. The abstract and described strategy indicate that the result is obtained by explicit construction of minimal generators whose exponent vectors control the appearance of new generators in successive powers. No equations reduce the claimed existence to a tautology or fitted input; the construction is presented as independent of the target sign sequence. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are referenced in the provided text. The unification of known examples is a consequence rather than a definitional reduction. The derivation is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard facts on minimal generators of powers of graded ideals and eventual increase of μ(I^k)
- domain assumption Properties of monomial ideals in K[x,y] allow independent control of initial differences via choice of generators
Reference graph
Works this paper leans on
-
[1]
Abbott, A
J. Abbott, A. M. Bigatti, L. Robbiano , CoCoA : a system for doing C omputations in C ommutative A lgebra , Available at http://cocoa.dima.unige.it/cocoalib
-
[2]
Abdolmaleki, J
R. Abdolmaleki, J. Herzog, Rashid Zaare-Nahandi , On the initial behavior of the number of generators of powers of monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie , Tome 63 (111) , No. 2. (2020), 119--129
2020
-
[3]
Abdolmaleki, S
R. Abdolmaleki, S. Kumashiro , Certain monomial ideals whose numbers of generators of powers descend. Arch. Math. (Basel) , 116 (2021), no. 6, 637--645
2021
-
[4]
Bruns, J
W. Bruns, J. Herzog , Cohen-Macaulay Rings, Cambridge University Press (1993)
1993
-
[5]
N. T. Cuong, P. H. Quy, H. L. Truong , On the index of reducibility in Noetherian modules, J. Pure Appl. Algebra , 219 (2015), 4510--4520
2015
-
[6]
Eliahou, J
S. Eliahou, J. Herzog, M. M. Saem , Monomial ideals with tiny squares, J. Algebra , 514 (2018), 99--112
2018
-
[7]
Gasanova , Monomial ideals with arbitrarily high tiny powers in any number of variables, Comm
O. Gasanova , Monomial ideals with arbitrarily high tiny powers in any number of variables, Comm. Algebra , 48 (2020), No. 11, 4824--4831
2020
discussion (0)
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