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Computation of dominant ideals

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Pith's one-line read New enumeration formulas count dominant monomial ideals with a fixed least common multiple in up to five variables, and simulations estimate how often they occur in random models.

desk verdict First closed-form counts of dominant ideals with fixed lcm up to five variables, with a clean associated-prime theorem, but the five-variable proof needs a completeness argument before the headline formula is trusted. read the letter →

arxiv 2504.12146 v1 pith:UC5DM2Z4 submitted 2025-04-16 math.AC math.CO

Anna Maria Bigatti , Nursel Erey , Selvi Kara , Augustine O'Keefe , Sonja Petrović , Pierpaola Santarsiero , Janet Striuli This is my paper · ORCID
classification math.ACmath.CO
keywords dominantidealsdeterminingmodelmonomialadoptallowingalternative
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

A monomial ideal is a set of monomials (products of variables) closed under adding multiples. It is called dominant when each of its minimal generators has a variable in which its exponent is strictly larger than that of every other generator. Dominant ideals are special because the Taylor resolution, a standard way to build free resolutions, is minimal exactly for them (a theorem of Alesandroni). So knowing how common dominant ideals are tells us how often the Taylor resolution is already as small as possible.

The paper's first contribution is combinatorial. Fixing the least common multiple of the generators, the authors count dominant ideals in two, three, four, and five variables. The counts are explicit polynomial formulas in the exponents of the fixed lcm. The key idea is the footprint of a generator: the list of variables in which it does not have the maximum exponent. Dominance forces the footprints to take only a few shapes, and the authors enumerate the shapes and multiply the number of choices for each exponent. The formulas were checked with the CoCoA computer algebra system, and the paper also describes an efficient algorithm for listing all dominant ideals with a given lcm.

The second contribution is probabilistic. Using the Erdős-Rényi type model of random monomial ideals, the authors simulate samples and plot the frequency of dominant ideals as the probability of selecting monomials varies. They find that dominance is very rare for moderate selection probabilities, and they propose a conjecture about the threshold in the homogeneous case. The simulations are exploratory: sample sizes are 50 to 1000, and no confidence intervals are reported, so the trends are indicative rather than proven.

Extended reading notes

Core claim

The number of dominant monomial ideals in five variables with a fixed lcm m=x1^{m1}...x5^{m5} is given by the closed-form formula in Theorem 3.12, obtained by a footprint partition whose cases the proof asserts to be exhaustive. Alongside it, Theorem 2.1 asserts that an associated prime of height k exists iff there is a dominating set of size k satisfying the stated covering property.

Load-bearing premise

The proof of Theorem 3.12 assumes the footprint families it lists are exhaustive for five variables; the text says 'Since we listed all possible cases this concludes the proof' without a separate completeness argument, so any omitted configuration of maximal and non-maximal exponents would change the count. The proof also assumes each exponent range in the listed cases yields distinct minimal generating sets with the prescribed lcm.

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Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The theorems use standard facts about monomial ideals and the paper's own footprint case analysis; the probabilistic part relies on an existing random model. No free parameters are fitted into the enumeration formulas; the only data-derived constants are thresholds read from simulations. No new entities are introduced.

free parameters (2)
  • probability threshold 0.1 in Conjecture 4.1 = 0.1 (reader-observed from simulations)
    The conjecture states that for n=3, as d grows, homogeneous ER random ideals are dominant with probability tending to 0 for p>0.1; this threshold is taken from the plotted simulation data, not derived from theory.
  • sampling probability grid midpoints (d^{-i}-d^{-(i-1)})/2 = varies with d and i
    These p values are chosen by hand to aim for samples of each Krull dimension in Figure 4 and Figure 5; they affect the plots but are not part of the theorems.
assumptions (6)
  • standard math Associated primes of monomial ideals are generated by subsets of variables (Herzog-Hibi, Corollary 1.3.9).
    Invoked in the proof of Theorem 2.1 to identify the associated prime with (x1,...,xk) and to argue the reverse containment by contradiction.
  • standard math For an associated prime P of a monomial ideal I, P = I:(v) for some monomial v, and the minimal generators of I:(v) are {u_j/gcd(u_j,v) : u_j in G(I)} (Herzog-Hibi, Proposition 1.2.2 and Corollary 1.3.10).
    Used in the (2)=>(1) direction of Theorem 2.1 to build the dominating set L from the colon ideal.
  • standard math Auslander-Buchsbaum formula and Hilbert Syzygy theorem: pd(S/I)=n iff depth(S/I)=0 iff the maximal ideal is associated to S/I.
    Used in Corollary 2.2 to connect Theorem 2.1 with Alesandroni's characterization of maximal projective dimension.
  • standard math Alesandroni's theorem: dominant monomial ideals are precisely those for which the Taylor resolution is minimal (Ale17).
    Used in the introduction to motivate why dominance is worth counting; not used in the proofs of the new formulas.
  • domain assumption The Erdős-Rényi type model from De Loera et al. is an appropriate model for random monomial ideals, and the graded and fixed-generator variants are suitable for studying dominance.
    Section 4 adopts these models as the sampling distribution and uses them to draw the empirical conclusions and Conjecture 4.1.
  • ad hoc to paper The footprint partition in Theorem 3.12 is exhaustive and each listed range of exponents produces distinct minimal generating sets with the fixed lcm.
    The proof of Theorem 3.12 states 'Since we listed all possible cases this concludes the proof' without a separate completeness argument; the formula depends on this case analysis being complete.

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Pith. "Pith review of Computation of dominant ideals." pith.science (2026). https://pith.science/paper/UC5DM2Z4

@misc{pith2026250412146,
  author       = {Pith},
  title        = {Pith review of: Computation of dominant ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UC5DM2Z4}},
  note         = {Machine review of arXiv:2504.12146}
}
read the original abstract

We consider the problem of determining whether a monomial ideal is dominant. This property is critical for determining for which monomial ideals the Taylor resolution is minimal. We first analyze dominant ideals with a fixed least common multiple of generators using combinatorial methods. Then, we adopt a probabilistic approach via the \er\ type model, examining both homogeneous and non-homogeneous cases. This model offers an efficient alternative to exhaustive enumeration, allowing the study of dominance through small random samples, even in high-dimensional settings.

Figures

Figures reproduced from arXiv: 2504.12146 by the authors.

Figure 1
Figure 1. Taylor complex (left) and Scarf complex (right) of I = (x1x2, x2x3, x1x3). The Taylor resolution of I ⊂ S = K[x1, x2, x3] being S 1 ( x 2 1 x1x3 x2x3 ) ←−−−−−−−−−−− S 3   −x3 −x2x3 0 x1 0 −x2 0 x 2 1 x1   ←−−−−−−−−−−−−− S 3  x2 −1 x1  ←−−−− S 1 . Although the non-minimality of the Taylor resolution is widely acknowledged, its extent has largely been discussed qualitatively rather than quantitatively. One of th… view at source ↗
Figure 4
Figure 4. shows the simulation results. Appendix A contains code that can be used to generate the data that produced these plots [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Frequency of dominant random homogeneous monomial ideals in IGr(n, d, p) for n = 3, d = 3, . . . , 12 and the nonzero probabilities taking values in n d ℓ−d ℓ−1 2 , ℓ = 2, . . . , no ∪  1 9 , 1 8 , . . . , 1 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figures from the paper (2 more)
Figure 6
Figure 6. Figure 6: Frequencies of observing a particular number of minimal generators for the ideals in the data set displayed in [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Proportion of dominant ideals in a sample of random monomial ideals. Each data point on each graph represents one sample of size 1000, for a fixed number of variables n, fixed generator degree D, and fixed number of generators [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

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