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arxiv: 2604.24523 · v1 · submitted 2026-04-27 · 🧮 math.AG

Denef-Loeser zeta functions of suspensions and L\^e-Yomdin singularities

Enrique Artal Bartolo , Pedro D. Gonz\'alez P\'erez , Manuel Gonz\'alez Villa , Edwin Le\'on Cardenal This is my paper

Pith reviewed 2026-05-08 01:48 UTC · model grok-4.3

classification 🧮 math.AG
keywords zeta functionssuspensionssingularitiesholomorphy conjecturemonodromy conjectureLê-Yomdin singularitiesmotivic zeta functionstopological zeta functions
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The pith

New formulas for zeta functions of suspensions prove the holomorphy conjecture for plane curve singularities and the holomorphy and monodromy conjectures for Lê-Yomdin surface singularities.

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

This paper derives general expressions for the motivic and topological zeta functions of a family of hypersurfaces that includes suspensions by an arbitrary number of points and handles arbitrary twisting parameters. These expressions generalize earlier formulas restricted to two-point suspensions and feature coefficients given by values of the Jordan totient function drawn from auxiliary lower-dimensional hypersurfaces. Using the new formulas, the authors establish the holomorphy conjecture for suspensions of plane curve singularities together with both the holomorphy and monodromy conjectures for Lê-Yomdin singularities of surfaces. A reader cares because the conjectures tie the locations of poles in zeta functions to concrete geometric features of the singularities, allowing algebraic data to predict analytic and topological behavior.

Core claim

The authors provide new formulas for the motivic and topological zeta functions of suspensions of hypersurfaces by an arbitrary number of points; these formulas are more general than Thom-Sebastiani type and apply for arbitrary values of the twisting parameter. The formulas involve the appearance of values of the Jordan totient function as coefficients in the zeta functions of certain auxiliary hypersurfaces of smaller dimension. With these expressions the paper proves the holomorphy conjecture for suspensions of plane curve singularities and proves both the holomorphy and monodromy conjectures for Lê-Yomdin singularities of surfaces.

What carries the argument

General formulas for the motivic and topological zeta functions of suspensions by an arbitrary number of points, which incorporate coefficients from the Jordan totient function evaluated on auxiliary lower-dimensional hypersurfaces.

If this is right

  • The zeta functions of these singularities reduce explicitly to data from lower-dimensional auxiliary hypersurfaces.
  • The poles of the zeta functions are located exactly where the geometry of the singularity predicts.
  • The holomorphy conjecture is settled for all suspensions of plane curve singularities.
  • Both the holomorphy and monodromy conjectures are settled for all Lê-Yomdin singularities of surfaces.

Where Pith is reading between the lines

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

  • The same reduction technique might extend to prove analogous conjectures for higher-dimensional Lê-Yomdin singularities.
  • Explicit computation of the Jordan totient coefficients for low-dimensional examples could produce new numerical checks of the formulas.
  • The approach may connect to other invariants such as the spectrum or the Milnor fiber cohomology.

Load-bearing premise

The new general formulas for zeta functions of suspensions by an arbitrary number of points hold for the stated family of hypersurfaces and correctly capture the twisting parameter for all values.

What would settle it

A concrete suspension of a plane curve singularity for which a pole of the zeta function violates the holomorphy conjecture, or an explicit hypersurface where the derived zeta-function formula fails to match direct computation, would disprove the central claim.

Figures

Figures reproduced from arXiv: 2604.24523 by Edwin Le\'on Cardenal, Enrique Artal Bartolo, Manuel Gonz\'alez Villa, Pedro D. Gonz\'alez P\'erez.

Figure 2
Figure 2. Figure 2: Resolution graph of (y 2 − x 3 ) 3 − x 6y 3 = 0. Lemma 6.12. The set of f-bad numbers is E ord f \ E ord f+z 2 . Proof. Denote by Bf ⊂ Z the set of f-bad numbers associated with f. Let us start with the inclusion Bf ⊂ E ord f \ E ord f+z 2 . Let ℓ be f-bad. From (B2), we have that ℓ ∈ E ord f . Let us prove by contradiction that ℓ /∈ E ord f+z 2 . If ℓ ∈ E ord f+z 2 , let d ∈ Eord f+z 2 such that ℓ | d. No… view at source ↗
Figure 3
Figure 3. Figure 3: Dual resolution graph of C = C1 with the multiplicities of the relevant exceptional components. Proof. Let C = r1C1+· · ·+rlCl be a decomposition of (C, 0) into locally irreducible components. Assume that all the eigenvalues of (C, 0) ⊂ (C 2 , 0) are bad. Since 1 is always an eigenvalue when l > 1, we can assume that there is only one irreducible component or branch, i.e, C = r1C1. If r1 > 2, then C has an… view at source ↗
Figure 4
Figure 4. Figure 4: Expected and final behavior around E0. E0 pi qi F i E0 pi Ei F i E0 pi E′ 0 p ′ i p ′ 5−i E′ 1 E′ 1 2 = −2 Ei ⫋ F i view at source ↗
Figure 5
Figure 5. Figure 5: Strata associated to pi . (S1) F i intersects the strict transform and has no branching component. The component F i is a bamboo, and we denote by qi the stratum in the strict transform. By (6.2), we have that F i ⊂ E( ℓ 2 ) , and then F i ⊂ C. (S2) F i does not intersect the strict transform and has no branching component. The com￾ponent Fi is a bamboo, and we denote by Ei its last exceptional component. … view at source ↗
Figure 6
Figure 6. Figure 6: Kashiwara’s pencils of types Ia, Ib, and II. Applying the valuation NEs , for an exceptional divisor Es of πγ corresponding to the left of the dicritical divisor E, to the divisors of (8.2), we have that NEs (F1) = p2NEs (Cp1 ) − p1NEs (Cp2 ) > 0 and NEs (F2) = 0. (8.3) And similar formulas hold for an exceptional divisor Es of πγ corresponding to the right of the dicritical divisor E. Notice also that NEs… view at source ↗
Figure 7
Figure 7. Figure 7: Pencils of quartics of type Ib. minimal resolution of Cm) is a branching component. Since ˜νi = νi − 3 m Ni , then − 3 m is a pole of Ztop(fq, s)0, and Cm is not a bad divisor. □ Remark 8.23. What happens if Cm does not contain all the special fibers? In this case, this special fiber is the image by π1 of the strict transform of EP , which has ˜νP = 0, i.e., it is not in the support of the canonical diviso… view at source ↗
Figure 8
Figure 8. Figure 8: Pencils of sextics with two Puiseux pairs of type Ia. Even though −1 appears as a coefficient, R(Cm) = 0, since the blow-ups from the ruled surface do not change the residue. Note that in this case ∆(τ ) has roots of order m even if it is not needed for the monodromy conjecture. This is a non-residual curve and in this case the eigenvalue associated to − e m is not of order m but of order m 3 = 2n. Note th… view at source ↗
Figure 9
Figure 9. Figure 9: Pencil of curves of degree 10 The divisors Γ∗ (γ(C2)) = E1 +E6 + 2E2 + 3E3 + 4E4 + 5(E5 +C2) and Γ∗ (γ(C5)) = E9 +E7 + 2(E8+C5) are fibers of the ruling defined by the pencil. We can blow-down these fibers such that the image of E6, E9 become fibers. In the ruling, we have as divisor E6+E9+E10+G1+· · ·+Gn. There are four choices for the curve Cm. [II]: For Cm = G1 + · · · + Gn, i.e., m = 10n and r = n, the… view at source ↗
read the original abstract

The holomorphy conjecture for suspensions of plane curve singularities and the holomorphy and monodromy conjectures for L\^e-Yomdin singularities of surfaces are proved. The first part of this paper provides formul{\ae} for the motivic and topological zeta functions for a family of hypersurfaces, including the suspensions by an arbitrary number of points and which are more general than Thom-Sebastiani type. These formulae generalize and are inspired by the description of the topological and the 2-twisted topological zeta functions of suspensions by 2 points of hypersurfaces, due to the first named author, Cassou-Nogu\`es, Luengo and Melle. The new general formul{\ae} deal with arbitrary values of the twisting parameter. An interesting feature of these general formul{\ae} is the appearance of values of the Jordan's totient function as coefficients of the topological and the twisted topological zeta functions of some auxiliary hypersurfaces of smaller dimension.

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

2 major / 1 minor

Summary. The paper derives new formulas for the motivic and topological Denef-Loeser zeta functions of a family of hypersurfaces that includes suspensions by an arbitrary number of points and is strictly larger than Thom-Sebastiani suspensions. These formulas generalize the 2-point results of Cassou-Noguès–Luengo–Melle to arbitrary twisting parameters and feature coefficients given by values of the Jordan totient function applied to auxiliary lower-dimensional zeta functions. The formulas are then substituted into generating functions to prove the holomorphy conjecture for suspensions of plane curve singularities and both the holomorphy and monodromy conjectures for Lê-Yomdin singularities of surfaces.

Significance. If the general zeta-function formulas hold, the work resolves longstanding conjectures on the location of poles of zeta functions and their compatibility with monodromy eigenvalues for these classes of singularities. The explicit appearance of Jordan totient values as coefficients and the extension to arbitrary suspension points constitute a technical contribution to the computation of motivic invariants beyond the classical Thom-Sebastiani setting.

major comments (2)
  1. [First part (general formulas for motivic and topological zeta functions)] The proofs of both the holomorphy conjecture (for suspensions of plane curves) and the holomorphy/monodromy conjectures (for Lê-Yomdin surfaces) rest entirely on substituting the new general zeta formulas into the relevant generating functions. The manuscript presents these formulas as an extension of the 2-point case, but supplies no independent verification—such as an explicit Denef–Loeser computation or computer-assisted check for a concrete 3-point suspension example—that the twisting-parameter dependence and the family of admissible hypersurfaces are correctly captured for n > 2.
  2. [Derivation of the general formulas and the substitution into the conjectures] The claim that the new formulas hold for a family strictly larger than Thom-Sebastiani suspensions and for every value of the twisting parameter is load-bearing; if the extension introduces hidden restrictions or fails to preserve the correct twisting dependence, the substitution step does not establish the conjectures. No direct comparison with known 2-point formulas is given to confirm reduction.
minor comments (1)
  1. [Section introducing the general zeta formulas] Notation for the twisting parameter and the auxiliary hypersurfaces should be introduced with explicit definitions before their first use in the formulas.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We respond point by point to the major remarks below.

read point-by-point responses

  1. Referee: [First part (general formulas for motivic and topological zeta functions)] The proofs of both the holomorphy conjecture (for suspensions of plane curves) and the holomorphy/monodromy conjectures (for Lê-Yomdin surfaces) rest entirely on substituting the new general zeta formulas into the relevant generating functions. The manuscript presents these formulas as an extension of the 2-point case, but supplies no independent verification—such as an explicit Denef–Loeser computation or computer-assisted check for a concrete 3-point suspension example—that the twisting-parameter dependence and the family of admissible hypersurfaces are correctly captured for n > 2.

    Authors: The general formulas are obtained in Section 2 by a direct computation of the motivic measure on the arc space, following the same strategy as the 2-point case but without any inductive step or restriction that would invalidate the argument for n > 2. The twisting parameter enters the exponent of the integral in a uniform way that is independent of the number of suspension points. Nevertheless, we agree that an explicit low-dimensional check would improve the exposition. In the revised version we will insert, after the statement of the main formula, a short explicit computation for the 3-point suspension of the plane curve singularity defined by x² + y³ = 0, comparing the output of the new formula with the value obtained from the original Denef–Loeser definition via a direct resolution. revision: yes

  2. Referee: [Derivation of the general formulas and the substitution into the conjectures] The claim that the new formulas hold for a family strictly larger than Thom-Sebastiani suspensions and for every value of the twisting parameter is load-bearing; if the extension introduces hidden restrictions or fails to preserve the correct twisting dependence, the substitution step does not establish the conjectures. No direct comparison with known 2-point formulas is given to confirm reduction.

    Authors: The construction in the paper allows hypersurfaces whose equation is not a sum of two functions in disjoint variables, hence strictly larger than the Thom–Sebastiani class; the motivic integral is performed on the full arc space without assuming such a decomposition. The twisting parameter appears explicitly in the definition of the zeta function and is carried through the computation without additional constraints. While the text asserts that the formulas recover the 2-point results of Cassou-Noguès–Luengo–Melle, we acknowledge that a side-by-side reduction is not displayed. We will add a dedicated remark (new Remark 2.6) that sets the number of points to 2 and the twisting parameter to 2, substitutes the Jordan totient values, and verifies term-by-term agreement with the earlier formulas. revision: partial

Circularity Check

0 steps flagged

New generalized zeta formulas derived for arbitrary suspensions and used to prove conjectures without reduction to prior inputs

full rationale

The paper states that it provides new formulas for motivic and topological zeta functions of a family of hypersurfaces that includes suspensions by an arbitrary number of points and is more general than Thom-Sebastiani type. These formulas are described as generalizing the 2-point case from prior work by the first author and collaborators, with the new versions handling arbitrary twisting parameters and featuring Jordan totient coefficients. The holomorphy and monodromy conjectures are then proved by substituting these formulas into the relevant generating functions. No equation or step in the provided description reduces the new formulas tautologically to the 2-point inputs by definition, fitting, or self-citation chain; the extension to arbitrary points and twisting introduces independent content. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no concrete list of free parameters, axioms, or invented entities; the work operates within standard motivic integration and singularity theory.

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