The strong monodromy conjecture for hyperplane arrangements

Lei Wu

arxiv: 2605.25335 · v2 · pith:UBKGEZD2new · submitted 2026-05-25 · 🧮 math.AG

The strong monodromy conjecture for hyperplane arrangements

Lei Wu This is my paper

Pith reviewed 2026-06-29 20:56 UTC · model grok-4.3

classification 🧮 math.AG

keywords hyperplane arrangementsb-functionstrong monodromy conjectureBudur-Mustaţă-Teitler conjectureBernstein-Sato polynomialtopological zeta functionalgebraic singularities

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The pith

−n/d is a root of the b-function for any irreducible essential central hyperplane arrangement of degree d in C^n

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

The paper proves that −n/d is a root of the b-function of an irreducible essential and central hyperplane arrangement f of degree d on C^n. This confirms the Budur-Mustaţă-Teitler conjecture and thereby establishes the strong monodromy conjecture for all complex hyperplane arrangements. A sympathetic reader cares because the result connects the algebraic roots of the b-function to the expected poles of the topological zeta function for this important class of hypersurface singularities. The argument applies the stated restrictions on the arrangements to locate the root.

Core claim

In this article, we prove the strong monodromy conjecture for complex hyperplane arrangements by proving a conjecture of Budur, Mustaţă and Teitler that −n/d is a root of the b-function of an irreducible essential and central hyperplane arrangement f of degree d on C^n.

What carries the argument

The b-function of the hyperplane arrangement polynomial f, shown to have −n/d among its roots under the given conditions on the arrangement.

Load-bearing premise

The hyperplane arrangements are irreducible, essential, and central.

What would settle it

An explicit computation of the b-function for one irreducible essential central hyperplane arrangement in which −n/d fails to be a root.

read the original abstract

In this article, we prove the strong monodromy conjecture for complex hyperplane arrangements by proving a conjecture of Budur, Musta\c t\u a and Teitler that $-n/d$ is a root of the $b$-function of an irreducible essential and central hyperplane arrangement $f$ of degree $d$ on $\C^n$.

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.

Desk Editor's Note private letter to a colleague

Wu proves the Budur-Mustaţă-Teitler root statement for irreducible essential central hyperplane arrangements, which settles the strong monodromy conjecture in this case.

read the letter

The paper shows that -n/d is a root of the b-function for an irreducible essential central hyperplane arrangement f of degree d in C^n. This directly gives the strong monodromy conjecture for these arrangements.

The new content is the proof of that root property, which had been conjectured but not established in the referenced work by Budur, Mustaţă and Teitler. The statement is scoped exactly to the cases where the conjecture applies, so the implication to the monodromy side is clean.

It handles a standard testing ground in singularity theory without overclaiming the scope. The conditions (irreducible, essential, central) are stated up front and match the conjecture's hypotheses.

The soft spot is that the abstract supplies no sketch of the argument, so the actual derivation steps, any use of prior results on arrangements, and checks for edge cases have to be verified in the body. If the proof is direct and avoids circular reductions, the result stands; otherwise the claim rests on details not visible here. No internal contradiction appears in the stated claim itself.

This is for people working on b-functions, monodromy, and hyperplane singularities. A reader already following that literature will get the value of the resolved case.

It deserves a serious referee because it claims to close a known conjecture with a precise, falsifiable statement.

Referee Report

1 major / 0 minor

Summary. The paper claims to prove the strong monodromy conjecture for complex hyperplane arrangements by proving the Budur-Mustaţă-Teitler conjecture: that −n/d is a root of the b-function of an irreducible essential and central hyperplane arrangement f of degree d on C^n.

Significance. If the asserted proof is correct, the result would resolve the strong monodromy conjecture in the setting of hyperplane arrangements, confirming a specific root of the Bernstein-Sato polynomial under the stated hypotheses and thereby settling an open question in singularity theory.

major comments (1)

[Abstract] Abstract: the manuscript asserts that a complete proof of the Budur-Mustaţă-Teitler conjecture (and hence the strong monodromy conjecture) is given, yet supplies no derivation steps, lemmas, or verification details, rendering the central claim impossible to check for mathematical soundness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the manuscript asserts that a complete proof of the Budur-Mustaţă-Teitler conjecture (and hence the strong monodromy conjecture) is given, yet supplies no derivation steps, lemmas, or verification details, rendering the central claim impossible to check for mathematical soundness.

Authors: The abstract is a concise summary of the main result, as is standard. The full manuscript contains the complete proof of the Budur-Mustaţă-Teitler conjecture (hence the strong monodromy conjecture), including all derivation steps, lemmas, and verification details needed to check the argument. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a direct mathematical proof that −n/d is a root of the Bernstein-Sato polynomial b_f(s) for irreducible essential central hyperplane arrangements of degree d, which in turn implies the strong monodromy conjecture. This is framed as a proof of the independent Budur-Mustaţă-Teitler conjecture under precisely the stated technical hypotheses, without any reduction of the central claim to fitted parameters, self-definitional loops, or load-bearing self-citations that themselves rest on unverified assumptions. The derivation chain is therefore self-contained as a standard mathematical argument against external benchmarks, with no steps that collapse by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard background from D-module theory and singularity theory for hyperplane arrangements; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of the Bernstein-Sato b-function and monodromy for hypersurface singularities in algebraic geometry
The paper invokes the existing theory of b-functions and the Budur-Mustaţă-Teitler conjecture as background.

pith-pipeline@v0.9.1-grok · 5561 in / 1196 out tokens · 30564 ms · 2026-06-29T20:56:26.661284+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 3 canonical work pages · 1 internal anchor

[1]

Part 1, 1--50

Alexander Beilinson and Joseph Bernstein, A proof of J antzen conjectures , Advances in Soviet mathematics 16 (1993), no. Part 1, 1--50

1993
[2]

2, 319--327

Alexander Beilinson and Dennis Gaitsgory, A corollary of the b-function lemma, Selecta Mathematica 18 (2012), no. 2, 319--327

2012
[3]

247, Springer Science & Business Media, 2013

Jan-Erik Bj \"o rk, Analytic D -modules and applications , vol. 247, Springer Science & Business Media, 2013

2013
[4]

Dedicata 153 (2011), 131--137

Nero Budur, Mircea Musta t a , and Zach Teitler, The monodromy conjecture for hyperplane arrangements, Geom. Dedicata 153 (2011), 131--137. 2819667

2011
[5]

Nero Budur, Morihiko Saito, and Sergey Yuzvinsky, On the local zeta functions and the b -functions of certain hyperplane arrangements , J. Lond. Math. Soc. (2) 84 (2011), no. 3, 631--648, With an appendix by Willem Veys. 2855794

2011
[6]

2, 549--603 (en)

Nero Budur, Bernstein- S ato ideals and local systems , Annales de l'Institut Fourier 65 (2015), no. 2, 549--603 (en)

2015
[7]

1, 45--72

Nero Budur, Robin van der Veer, Lei Wu, and Peng Zhou, Zero loci of B ernstein-- S ato ideals , Inventiones mathematicae 225 (2021), no. 1, 45--72

2021
[8]

Asilata Bapat and Robin Walters, The strong topological monodromy conjecture for W eyl hyperplane arrangements , Math. Res. Lett. 24 (2017), no. 4, 947--954. 3723798

2017
[9]

3, 459--494

Corrado De Concini and Claudio Procesi, Wonderful models of subspace arrangements, Selecta Mathematica 1 (1995), no. 3, 459--494

1995
[10]

4, 705--720

Jan Denef and Fran c ois Loeser, Caract \'e ristiques d' E uler- P oincar \'e , fonctions z \^e ta locales et modifications analytiques , Journal of the American Mathematical Society 5 (1992), no. 4, 705--720

1992
[11]

4, 2331--2367

Graham Denham, Alexander I Suciu, and Sergey Yuzvinsky, Abelian duality and propagation of resonance, Selecta Mathematica 23 (2017), no. 4, 2331--2367

2017
[12]

Dougal Davis and Ruijie Yang, Multivariate V -filtrations and the strong monodromy conjecture for hyperplane arrangements , arXiv preprint arXiv:2605.25314 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[13]

150, Springer-Verlag, New York, 1995

David Eisenbud, Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995

1995
[14]

Victor Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), no. 2, 327--402. 833194

1986
[15]

236, Birkh\" a user Boston, Inc., Boston, MA, 2008, Translated from the 1995 Japanese edition by Takeuchi

Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, D -modules, perverse sheaves, and representation theory , Progress in Mathematics, vol. 236, Birkh\" a user Boston, Inc., Boston, MA, 2008, Translated from the 1995 Japanese edition by Takeuchi. 2357361

2008
[16]

Masaki Kashiwara, B-functions and holonomic systems, Inventiones Mathematicae 38 (1976), 33--53

1976
[17]

1, 283--304

Ben Lichtin, Poles of |f (z, w)|^ 2s and roots of the b-function , Arkiv f \"o r Matematik 27 (1989), no. 1, 283--304

1989
[18]

Luis Narv \'a ez Macarro, A duality approach to the symmetry of B ernstein-- S ato polynomials of free divisors , Advances in Mathematics 281 (2015), 1242--1273

2015
[19]

, Rendiconti del Seminario Matematico della Universita di Padova 150 (2023), 81--125

Philippe Maisonobe, Filtration relative, l'id \'e al de bernstein et ses pentes. , Rendiconti del Seminario Matematico della Universita di Padova 150 (2023), 81--125

2023
[20]

R elative finiteness and duality , Ast \'e risque (1994), no

Pierre Schapira and Jean-Pierre Schneiders, Elliptic pairs I . R elative finiteness and duality , Ast \'e risque (1994), no. 224, 5--60

1994
[21]

Pure Appl

Vadim Schechtman, Hiroaki Terao, and Alexander Varchenko, Local systems over complements of hyperplanes and the K ac- K azhdan conditions for singular vectors , J. Pure Appl. Algebra 100 (1995), 93--102

1995
[22]

Quan Shi and Huaiqing Zuo, Variation of archimedean zeta function and n/d -conjecture for generic multiplicities, arXiv preprint arXiv:2411.00757 (2024)

work page arXiv 2024
[23]

5, 1575--1579

Zach Teitler, A note on mustata’s computation of multiplier ideals of hyperplane arrangements, Proceedings of the American Mathematical Society 136 (2008), no. 5, 1575--1579

2008
[24]

Uli Walther, The J acobian module, the M ilnor fiber, and the D -module generated by f^s , Invent. Math. 207 (2017), no. 3, 1239--1287. 3608290

2017
[25]

Lei Wu, On the comparison of nearby cycles via b -functions, J. Singul. 23 (2021), 92--106

2021
[26]

9, Cambridge University Press, 2021

Lei Wu and Peng Zhou, Log D -modules and index theorems , Forum of Mathematics, Sigma, vol. 9, Cambridge University Press, 2021

2021
[27]

Baiting Xie and Chenglong Yu, The n/d-conjecture for nonresonant hyperplane arrangements, arXiv preprint arXiv.2501.05189 (2025)

work page arXiv 2025

[1] [1]

Part 1, 1--50

Alexander Beilinson and Joseph Bernstein, A proof of J antzen conjectures , Advances in Soviet mathematics 16 (1993), no. Part 1, 1--50

1993

[2] [2]

2, 319--327

Alexander Beilinson and Dennis Gaitsgory, A corollary of the b-function lemma, Selecta Mathematica 18 (2012), no. 2, 319--327

2012

[3] [3]

247, Springer Science & Business Media, 2013

Jan-Erik Bj \"o rk, Analytic D -modules and applications , vol. 247, Springer Science & Business Media, 2013

2013

[4] [4]

Dedicata 153 (2011), 131--137

Nero Budur, Mircea Musta t a , and Zach Teitler, The monodromy conjecture for hyperplane arrangements, Geom. Dedicata 153 (2011), 131--137. 2819667

2011

[5] [5]

Nero Budur, Morihiko Saito, and Sergey Yuzvinsky, On the local zeta functions and the b -functions of certain hyperplane arrangements , J. Lond. Math. Soc. (2) 84 (2011), no. 3, 631--648, With an appendix by Willem Veys. 2855794

2011

[6] [6]

2, 549--603 (en)

Nero Budur, Bernstein- S ato ideals and local systems , Annales de l'Institut Fourier 65 (2015), no. 2, 549--603 (en)

2015

[7] [7]

1, 45--72

Nero Budur, Robin van der Veer, Lei Wu, and Peng Zhou, Zero loci of B ernstein-- S ato ideals , Inventiones mathematicae 225 (2021), no. 1, 45--72

2021

[8] [8]

Asilata Bapat and Robin Walters, The strong topological monodromy conjecture for W eyl hyperplane arrangements , Math. Res. Lett. 24 (2017), no. 4, 947--954. 3723798

2017

[9] [9]

3, 459--494

Corrado De Concini and Claudio Procesi, Wonderful models of subspace arrangements, Selecta Mathematica 1 (1995), no. 3, 459--494

1995

[10] [10]

4, 705--720

Jan Denef and Fran c ois Loeser, Caract \'e ristiques d' E uler- P oincar \'e , fonctions z \^e ta locales et modifications analytiques , Journal of the American Mathematical Society 5 (1992), no. 4, 705--720

1992

[11] [11]

4, 2331--2367

Graham Denham, Alexander I Suciu, and Sergey Yuzvinsky, Abelian duality and propagation of resonance, Selecta Mathematica 23 (2017), no. 4, 2331--2367

2017

[12] [12]

Dougal Davis and Ruijie Yang, Multivariate V -filtrations and the strong monodromy conjecture for hyperplane arrangements , arXiv preprint arXiv:2605.25314 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[13] [13]

150, Springer-Verlag, New York, 1995

David Eisenbud, Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995

1995

[14] [14]

Victor Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), no. 2, 327--402. 833194

1986

[15] [15]

236, Birkh\" a user Boston, Inc., Boston, MA, 2008, Translated from the 1995 Japanese edition by Takeuchi

Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, D -modules, perverse sheaves, and representation theory , Progress in Mathematics, vol. 236, Birkh\" a user Boston, Inc., Boston, MA, 2008, Translated from the 1995 Japanese edition by Takeuchi. 2357361

2008

[16] [16]

Masaki Kashiwara, B-functions and holonomic systems, Inventiones Mathematicae 38 (1976), 33--53

1976

[17] [17]

1, 283--304

Ben Lichtin, Poles of |f (z, w)|^ 2s and roots of the b-function , Arkiv f \"o r Matematik 27 (1989), no. 1, 283--304

1989

[18] [18]

Luis Narv \'a ez Macarro, A duality approach to the symmetry of B ernstein-- S ato polynomials of free divisors , Advances in Mathematics 281 (2015), 1242--1273

2015

[19] [19]

, Rendiconti del Seminario Matematico della Universita di Padova 150 (2023), 81--125

Philippe Maisonobe, Filtration relative, l'id \'e al de bernstein et ses pentes. , Rendiconti del Seminario Matematico della Universita di Padova 150 (2023), 81--125

2023

[20] [20]

R elative finiteness and duality , Ast \'e risque (1994), no

Pierre Schapira and Jean-Pierre Schneiders, Elliptic pairs I . R elative finiteness and duality , Ast \'e risque (1994), no. 224, 5--60

1994

[21] [21]

Pure Appl

Vadim Schechtman, Hiroaki Terao, and Alexander Varchenko, Local systems over complements of hyperplanes and the K ac- K azhdan conditions for singular vectors , J. Pure Appl. Algebra 100 (1995), 93--102

1995

[22] [22]

Quan Shi and Huaiqing Zuo, Variation of archimedean zeta function and n/d -conjecture for generic multiplicities, arXiv preprint arXiv:2411.00757 (2024)

work page arXiv 2024

[23] [23]

5, 1575--1579

Zach Teitler, A note on mustata’s computation of multiplier ideals of hyperplane arrangements, Proceedings of the American Mathematical Society 136 (2008), no. 5, 1575--1579

2008

[24] [24]

Uli Walther, The J acobian module, the M ilnor fiber, and the D -module generated by f^s , Invent. Math. 207 (2017), no. 3, 1239--1287. 3608290

2017

[25] [25]

Lei Wu, On the comparison of nearby cycles via b -functions, J. Singul. 23 (2021), 92--106

2021

[26] [26]

9, Cambridge University Press, 2021

Lei Wu and Peng Zhou, Log D -modules and index theorems , Forum of Mathematics, Sigma, vol. 9, Cambridge University Press, 2021

2021

[27] [27]

Baiting Xie and Chenglong Yu, The n/d-conjecture for nonresonant hyperplane arrangements, arXiv preprint arXiv.2501.05189 (2025)

work page arXiv 2025