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arxiv: 2602.24153 · v5 · submitted 2026-02-27 · 🧮 math.AG

Strong monodromy conjecture for defining polynomials of projective hypersurfaces having only weighted homogeneous isolated singularities

Morihiko Saito This is my paper

Pith reviewed 2026-05-15 18:41 UTC · model grok-4.3

classification 🧮 math.AG
keywords strong monodromy conjectureweighted homogeneous singularitiesprojective hypersurfacesBernstein-Sato polynomialtopological zeta functionNewton-nondegenerate polynomialsisolated singularities
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The pith

The strong monodromy conjecture follows from prior results for defining polynomials of projective hypersurfaces with weighted homogeneous isolated singularities in specified cases.

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

This paper shows that the strong monodromy conjecture for a defining polynomial f of a projective hypersurface Z follows from an earlier arXiv result when Z is a reduced curve or when the reduced hypersurface has only homogeneous isolated singularities with ambient dimension at least 4. The argument applies the Denef-Loeser formula for Newton-nondegenerate polynomials of three variables, together with known results on the conjecture in two variables, and relies on a cancellation in the formulas that prevents potential counterexamples from appearing. The paper also establishes a direct relation between the orders of poles in the topological zeta function and the multiplicities of roots of the Bernstein-Sato polynomial when Z has equimultiplicity and the singularities satisfy the weighted homogeneous condition in dimensions three or higher.

Core claim

In the case Z is a reduced curve or Z_red has only homogeneous isolated singularities with n at least 4, the strong monodromy conjecture for f follows from arxiv:1609.04801v1 using in the reduced curve case a formula of Denef and Loeser for Newton-nondegenerate polynomials of three variables together with known results about the strong monodromy conjecture in the two variable case. Here an amazing cancellation occurs so that possible counterexamples fail. The paper also shows the relation between the pole orders of topological zeta function and the root multiplicities of Bernstein-Sato polynomial in the case Z has equimultiplicity and Z_red has only weighted homogeneous singularities with n=

What carries the argument

Reduction of the strong monodromy conjecture to a prior theorem via the Denef-Loeser formula for Newton-nondegenerate polynomials, combined with cancellation in the expressions for pole orders and root multiplicities.

If this is right

  • The strong monodromy conjecture holds for every reduced curve whose singularities are weighted homogeneous and isolated.
  • The conjecture holds for hypersurfaces in projective space of dimension at least four whose reduced form has only homogeneous isolated singularities.
  • Pole orders of the topological zeta function equal specific root multiplicities of the Bernstein-Sato polynomial whenever Z has equimultiplicity and the reduced singularities are weighted homogeneous with n equal to three or homogeneous isolated with n greater than three.

Where Pith is reading between the lines

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

  • The same cancellation mechanism might permit extension of the conjecture to singularities that are not weighted homogeneous if analogous formulas can be derived.
  • Explicit computation of the topological zeta function for low-degree examples could verify the cancellation numerically.
  • The established link between zeta poles and Bernstein-Sato roots suggests a direct dictionary between topological and algebraic invariants for this class of singularities.

Load-bearing premise

The hypersurface Z or its reduction must have only weighted homogeneous isolated singularities for the reduction to the prior result to remain valid after the observed cancellation.

What would settle it

A concrete projective hypersurface with only weighted homogeneous isolated singularities whose defining polynomial has a pole order in the topological zeta function that does not satisfy the predicted relation to Bernstein-Sato root multiplicities.

read the original abstract

Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{\rm red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{\rm red}$ has only homogeneous isolated singularities with $n$ at least $4$, we show that the strong monodromy conjecture for a defining polynomial $f$ of $Z$ follows from arxiv:1609.04801v1 using in the reduced curve case a formula of Denef and Loeser for Newton-nondegenerate polynomials of three variables (which can be deduced in the applied case from the one for the two variable case) together with known results about the strong monodromy conjecture in the two variable case. Here an amazing cancellation occurs so that possible counterexamples fail. We also show the relation between the pole orders of topological zeta function and the root multiplicities of Bernstein-Sato polynomial in the case $Z$ has equimultiplicity and $Z_{\rm red}$ has only weighted homogeneous singularities with $n=3$ or $Z_{\rm red}$ has only homogeneous isolated singularities with $n>3$.

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

1 major / 1 minor

Summary. The manuscript claims that for a projective hypersurface Z whose reduced form Z_red has only weighted homogeneous isolated singularities, the strong monodromy conjecture for a defining polynomial f holds when Z is a reduced curve or when the singularities are homogeneous with n ≥ 4. This follows from the result in arXiv:1609.04801v1 by applying the Denef-Loeser formula for Newton-nondegenerate polynomials in three variables (which reduces to the two-variable case) together with known two-variable results; an 'amazing cancellation' is asserted to eliminate potential counterexamples. The paper also claims a relation between the pole orders of the topological zeta function and the root multiplicities of the Bernstein-Sato polynomial in the equimultiplicity case with n=3 (weighted homogeneous) or n>3 (homogeneous isolated singularities).

Significance. If the reduction and cancellation are verified, the result would provide a concrete advance on the strong monodromy conjecture for hypersurfaces with weighted homogeneous isolated singularities by reducing it to prior work and low-dimensional cases. The asserted cancellation phenomenon and the explicit link between topological zeta poles and Bernstein-Sato roots would be useful additions to the literature on singularity invariants.

major comments (1)
  1. The central claim rests on an 'amazing cancellation' that causes possible counterexamples to fail when reducing to arXiv:1609.04801v1 via the Denef-Loeser formula. Without the explicit computation of this cancellation (including how the three-variable formula is deduced from the two-variable case and how the reduction applies after cancellation), it is impossible to confirm that the argument is load-bearing and free of gaps.
minor comments (1)
  1. The abstract uses 'n at least 4' and 'n=3 or n>3' without stating the ambient dimension convention for the projective space P^{n-1}; a brief clarification of the indexing would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for greater explicitness in our treatment of the cancellation phenomenon. We agree that this aspect of the argument requires expansion to make the reduction fully transparent and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The central claim rests on an 'amazing cancellation' that causes possible counterexamples to fail when reducing to arXiv:1609.04801v1 via the Denef-Loeser formula. Without the explicit computation of this cancellation (including how the three-variable formula is deduced from the two-variable case and how the reduction applies after cancellation), it is impossible to confirm that the argument is load-bearing and free of gaps.

    Authors: We agree that the current presentation of the cancellation is insufficiently detailed for independent verification. In the revised manuscript we will add a self-contained subsection that (i) recalls the Denef-Loeser formula for Newton-nondegenerate polynomials in three variables and shows its reduction to the two-variable case under the weighted-homogeneous isolated-singularity hypothesis, and (ii) carries out the explicit term-by-term cancellation that eliminates the potential counterexamples when the result of arXiv:1609.04801v1 is applied. This expansion will not change the logical structure or the statements of the theorems, but will render the argument fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract states that the strong monodromy conjecture follows from a prior arXiv paper (arxiv:1609.04801v1) using additional known results and formulas. No internal equations or definitions are provided that reduce the claim to its own inputs by construction. The derivation is presented as a consequence of external prior work, with no evidence of self-referential logic within the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on the validity of the strong monodromy conjecture in the two-variable case and on the Denef-Loeser formula for Newton-nondegenerate polynomials; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The strong monodromy conjecture holds for the two-variable case and for Newton-nondegenerate polynomials of three variables via the Denef-Loeser formula
    Invoked explicitly to deduce the three-variable reduced-curve case from the two-variable case.
  • domain assumption Known results on the strong monodromy conjecture apply directly after the cancellation
    Used to complete the reduction for n at least 4 with homogeneous singularities.

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