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arxiv: 2212.09528 · v5 · submitted 2022-12-19 · 🧮 math.AC · math.CO

Polarizations of Artin monomial ideals

Gunnar Fl{\o}ystad , Ine Gabrielsen , Amir Mafi This is my paper

Pith reviewed 2026-05-24 10:15 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords polarizationsArtin monomial idealstriangulated ballsCohen-Macaulay subcomplexesAlexander dualitycellular resolutionssqueezed ballsjoined spheres
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The pith

Any polarization of an Artin monomial ideal defines a triangulated ball on the join of simplex boundaries.

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

The paper establishes that every polarization of an Artin monomial ideal containing (x1^a1, ..., xn^an) produces a full-dimensional triangulated ball on the sphere formed by joining the boundaries of simplices of dimensions a1-1 through an-1. This directly proves the stated conjecture. The work further shows that all full-dimensional Cohen-Macaulay subcomplexes of this joined sphere arise exactly from polarizations and that the resulting balls are constructible. The dual cell complex inside the product of the simplices supplies a cellular minimal free resolution of the Alexander dual ideal. Squeezed balls are recovered as special cases of these polarizations.

Core claim

We show that any polarization of an Artin monomial ideal defines a triangulated ball. Geometrically, polarizations of ideals containing (x1^a1, …, xn^an) define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions a1-1, ⋯, an-1. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible. Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions a1-1, ⋯, an-1. We prove that this cell complex gives cellular minimal free resolution of the Alexander dual ideal of the triangulated ball.

What carries the argument

Polarization of an Artin monomial ideal, realized geometrically as a full-dimensional triangulation of the join of simplex boundaries.

If this is right

  • The triangulated balls obtained this way are constructible.
  • The dual cell complex inside the product of simplices supplies a cellular minimal free resolution of the Alexander dual ideal.
  • When the product of simplices is a hypercube, the dual cell complexes classify all polarizations in a stated range of examples.
  • Kalai's squeezed balls arise directly as polarizations of Artin monomial ideals.

Where Pith is reading between the lines

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

  • The geometric correspondence supplies an explicit way to build cellular resolutions for Alexander duals of these monomial ideals.
  • The classification technique used in the hypercube case may extend to other products of simplices beyond the hypercube.
  • The recovery of squeezed balls suggests that other known families of triangulations could be recovered algebraically via polarizations.

Load-bearing premise

The algebraic definition of a polarization matches the geometric triangulation of the joined sphere without further restrictions.

What would settle it

An explicit polarization of some Artin monomial ideal whose associated complex is not a triangulated ball, or a full-dimensional Cohen-Macaulay subcomplex of the joined sphere that cannot be obtained from any polarization.

Figures

Figures reproduced from arXiv: 2212.09528 by Amir Mafi, Gunnar Fl{\o}ystad, Ine Gabrielsen.

Figure 1
Figure 1. Figure 1: Simplicial complexes of J1 and J2 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Triangulated balls defined by J1 and J2 The essential element in proving Theorem 4.2 is to show the following: Theorem 4.3 Any full-dimensional Cohen-Macaulay proper sub-complex on the simplicial sphere ∂∆(A•) is a constructible triangulated ball. In fact, such sub-complexes X are precisely those that via the Stanley-Reisner correspondence come from polarizations of Artin monomial ideals. As noted above (s… view at source ↗
Figure 3
Figure 3. Figure 3: Simplicial complex of ideal (xz, xw, yz, yw, zw) The associated cell complex C(T) will then consist of a square and two edges (in general a (d−1)-simplex is turned into a d-hypercube, when each αi = 2). The cell complex must be acyclic. This gives, up to isomorphism, the four possibilities for C(T) given in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Acyclic cell complexes of one square and two line seg￾ments These correspond to the four, up to isomorphism, possible polarizations of the Artin ideal j. They can be worked out to be: J1 = (x0x1, y0y1, z0z1, w0w1) + (x0z0, x0w0, y0z0, y0w0, z0w0) J2 = (x0x1, y0y1, z0z1, w0w1) + (x0z0, x0w0, y1z0, y0w0, z0w0) J3 = (x0x1, y0y1, z0z1, w0w1) + (x1z0, x0w0, y1z0, y0w0, z0w0) J4 = (x0x1, y0y1, z0z1, w0w1) + (x0z… view at source ↗
Figure 5
Figure 5. Figure 5: Triangulated ball of Example 3.7 Remark 3.9. The polarizations of powers of graded maximal ideals (x1, . . . , xn) m are classified by Almousa, Lohne and the first author in [3]. In particular polar￾izations of (x1, . . . , xn) 2 are in bijections with trees with n edges. These results in [3] are partially extended to strongly stable ideals generated in a single degree in Hao, Luo and Saint Rain [26, Prop.… view at source ↗
Figure 6
Figure 6. Figure 6: The square Π(A•), with monomial labels on vertices The augmented chain complex is k ← k 4 ← k 4 ← k, which has no homology, it is acyclic. Let S = k[a1, b1, a2, b2]. The rainbow ideal is I(T) = (a1a2, a1b2, b1a2, b1b2). The augmented free cellular complex is S d ← S(−2)4 ← S(−3)4 ← S(−4). The image of d is I(T), and in this case, omitting the first term, the complex is a minimal free resolution of I(T). In… view at source ↗
Figure 7
Figure 7. Figure 7: The cube with monomial labelling of vertices In the following, for a subset S of [n] write x(S) for the ideal (xi)i∈S and x 2 (S) = (x 2 i )i∈S. It is well known that the minimal primary decomposition of IX is (22) IX = \ facetF of X x(F c ), see [28, Lemma 1.5.4] or [39, Theorem 1.7]. Proposition 7.2. The minimal irreducible decomposition of j = m□ + j∆ is j = \ facetFofX (x(F c ) + x 2 (F)). In particula… view at source ↗
Figure 8
Figure 8. Figure 8: The cube with monomial labelling We write it somewhat more stylistic as in the right of [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The C(T) are trees with labellings bbb aaa bbb aaa [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The C(T)-trees with their embedding on cubes corresponding polarizations are then J1 = (xaxb, yayb, zazb) + (xaya, xaza, yaza) J2 = (xaxb, yayb, zazb) + (xayb, xazb, yazb) More simply, using Corollary 6.8 to obtain the polarization J2, amounts to observe for the tree to the right in [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Alternative labelling of a and b, with corresponding embedding on cube This gives a polarization isomorphic to J2: J3 = (xaxb, yayb, zazb) + (xbyb, xbza, yaza) [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Simplicial complex: A line segment and two isolated points In a polarization J(T), by Theorem 6.7 the cell complex C(T) will then have one square, corresponding to the line segment labelled xy, and two line segments corresponding to the points labelled z and w. The cell complex C(T) must also be r-acyclic. This gives, up to isomorphism, the four possibilities for C(T) given in [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 13
Figure 13. Figure 13: Restriction-acyclic C(T) with one square and two line segments Theorem 6.7 and Corollary 6.8 they are seen to be: J1 = (xaxb, yayb, zazb, wawb) + (xaza, xawa, yaza, yawa, zawa) J2 = (xaxb, yayb, zazb, wawb) + (xaza, xawa, ybza, yawa, zawa) J3 = (xaxb, yayb, zazb, wawb) + (xbza, xawa, ybza, yawa, zawa) J4 = (xaxb, yayb, zazb, wawb) + (xaza, xawa, yaza, yawa, zbwa) (Again for each variable, the labels a and… view at source ↗
Figure 14
Figure 14. Figure 14: Three line segments J(T), the cell complex C(T) will then have three squares, corresponding to the lines labelled xy, zw and tu. The cell complex C(T) must also be r-acyclic. By Corollary 6.9, each pair of squares can intersect in at most a point. This gives, up to isomorphism, the three possibilities for C(T) given in [PITH_FULL_IMAGE:figures/full_fig_p035_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Restriction-acyclic C(T) of three squares without edge intersections up to isomorphism, three polarizations of j. Using Theorem 6.7 and Corollary 6.8 they are seen to be (write M□ = (xaxb, yayb, zazb, wawb, tatb, uaub)) J1 = M□ + (xazb, xawb, yazb, yawb, xatb, xaua, yatb, yaua, zatb, zaua, wbtb, wbua) J2 = M□ + (xazb, xawb, yazb, yawb, xatb, xaub, yatb, yaub, zatb, zaub, watb, waub) J3 = M□ + (xaza, xawa,… view at source ↗
Figure 16
Figure 16. Figure 16: Two line segments and an isolated point In a polarization J(T), the cell complex C(T) will then have two squares, cor￾responding to the lines labelled xy and yz, and one line segment, labelled w. The cell complex C(T) must be r-acyclic. This gives, up to isomorphism, the two possibilities for C(T) given in [PITH_FULL_IMAGE:figures/full_fig_p036_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Restriction-acyclic C(T) with two adjacent squares and one line segment x z w y [PITH_FULL_IMAGE:figures/full_fig_p037_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The star with three edges lines labelled xy, xz and xw. The cell complex C(T) must be r-acyclic. This gives, up to isomorphism, the two possibilities for C(T) given in [PITH_FULL_IMAGE:figures/full_fig_p037_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Restriction-ayclic C(T) with three adjacent squares get, up to isomorphism, two polarizations of j. They are seen to be: J1 = (xaxb, yayb, zazb, wawb) + (yaza, yawa, zawa) J2 = (xaxb, yayb, zazb, wawb) + (yazb, yawb, zawb) Example 7.11. j = (x 2 , y2 , z2 , w2 , t2 ) + (xz, yw, zt, wx, ty). The simplicial complex X associated to j∆ is given in [PITH_FULL_IMAGE:figures/full_fig_p037_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The five-cycle t x z y x t w y z w x y w z t [PITH_FULL_IMAGE:figures/full_fig_p038_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Restriction-acyclic C(T) with five squares in a cycle is the standard polarization: J = (xaxb, yayb, zazb, wawb, tatb) + (xaza, yawa, zata, waxa, taya). In general, if j∆ corresponds to the simplicial complex which is a cycle graph, then j has only the standard polarization. This is a consequence of the following, which is an analog of [22, Corollary 2.3]. Proposition 7.12. Suppose j = m□ + j∆, where the … view at source ↗
Figure 22
Figure 22. Figure 22: Vertices correspond to generators of (x1, x2, x3) 3 D.Lu and Z.Wang, [34], extend this. They replace the down-set I of Hom(P, [n]) with larger classes of subsets of Hom(P, [n]). Polarizations of powers of maximal graded ideals (x1, . . . , xn) m are characterized in [3]. When n = 3, there is a nice visual procedure for this. Planar triangulations n = 3. The generators of (x1, x2, x3) m corresponds to the … view at source ↗
read the original abstract

We show that any polarization of an Artin monomial ideal defines a triangulated ball. This proves a conjecture of A.Almousa, H.Lohne and the first author. Geometrically, polarizations of ideals containing $(x_1^{a_1}, \ldots, x_n^{a_n})$ define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions $a_1-1, \cdots, a_n-1$. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible. Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions $a_1-1, \cdots a_n-1$. We prove that this cell complex gives cellular minimal free resolution of this of the Alexander dual ideal of the triangulated ball. When the product of simplices is a hypercube, using these dual cell complexes we classify in a range examples all polarizations of the Artin monomial ideal. We also show that the squeezed balls of G.Kalai \cite{Ka} derive from polarizations of Artin monomial ideals.

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 / 3 minor

Summary. The paper proves that every polarization of an Artin monomial ideal yields a triangulated ball on the join of boundaries of simplices of dimensions a1-1 to an-1, thereby establishing a conjecture of Almousa, Lohne, and the first author. It shows the converse (every full-dimensional Cohen-Macaulay subcomplex arises this way), proves the balls are constructible, constructs the dual cell complex inside the product of simplices, proves this complex supports the cellular minimal free resolution of the Alexander dual, classifies polarizations in a range of hypercube cases, and derives Kalai's squeezed balls from such polarizations.

Significance. If the central correspondence and resolution claims hold, the work supplies an explicit geometric model and cellular resolution construction for a broad class of monomial ideals, resolves an open conjecture, and connects algebraic polarizations to constructible balls and squeezed spheres. The explicit dual-cell-complex construction and the hypercube classification are concrete strengths that could be used for further computations or generalizations.

minor comments (3)
  1. [Abstract] Abstract: the sentence on the dual cell complex and Alexander dual resolution is compressed; a parenthetical reference to the relevant theorem number would improve readability.
  2. [Introduction] The classification statement for the hypercube case is qualified by 'in a range examples'; the precise range (e.g., specific values of a_i or dimension) should be stated explicitly in the introduction or the relevant theorem.
  3. [Introduction] The reference to Kalai's squeezed balls appears only in the abstract and the final sentence; a short comparison paragraph in the introduction would clarify how the polarization construction recovers or extends the earlier examples.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results and the recommendation for minor revision. The report accurately captures the main contributions, including the proof of the conjecture, the converse statement, constructibility, the dual cell complex construction supporting the cellular resolution of the Alexander dual, the classification in hypercube cases, and the derivation of Kalai's squeezed balls. Since no specific major comments are provided in the report, we have no individual points to rebut or revise at this stage. We will incorporate any minor editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result is a direct proof that polarizations of Artin monomial ideals yield triangulated balls, thereby establishing the cited conjecture of Almousa-Lohne-Fløystad. The abstract and structure indicate that the geometric correspondence, constructibility, dual cell complexes, and cellular resolutions are derived via explicit algebraic and combinatorial arguments within the manuscript itself. The self-citation merely identifies the target conjecture rather than supplying a load-bearing premise or uniqueness theorem; no definitions reduce to their own outputs, no fitted parameters are relabeled as predictions, and no ansatz is imported via prior self-work. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions from commutative algebra (monomial ideals, polarizations, Alexander duality) and combinatorial topology (simplicial complexes, Cohen-Macaulay property, constructible balls). No free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Standard properties of monomial ideals and their polarizations as defined in the literature on Artin ideals.
    Invoked throughout to connect algebraic polarizations to geometric triangulations.
  • standard math Properties of joins of simplex boundaries and Cohen-Macaulay subcomplexes.
    Used to characterize the spheres and balls in the geometric statements.

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