Arithmetic properties and zeros of the Bergman kernel on a class of quotient domains

Luke D. Edholm; Vikram T. Mathew

arxiv: 2505.20489 · v3 · pith:A4ZLCJTGnew · submitted 2025-05-26 · 🧮 math.CV

Arithmetic properties and zeros of the Bergman kernel on a class of quotient domains

Luke D. Edholm , Vikram T. Mathew This is my paper

Pith reviewed 2026-05-22 02:26 UTC · model grok-4.3

classification 🧮 math.CV

keywords Bergman kernelLu Qi-Keng problemquotient domainszeros of kernelsrational exponentsarithmetic propertiescomplex analysisReinhardt domains

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The pith

Explicit formula for Bergman kernel on H_γ shows zeros for sequences of rationals γ_j approaching 1.

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

The paper derives an explicit formula for the Bergman kernel on the domains H_γ = {|z1|^γ < |z2| < 1} when γ is a rational number m/n greater than 1. The formula depends on the arithmetic properties of this rational, which produces new symmetries and clarifies earlier work on these quotient domains. The authors then apply the formula to construct sequences of rationals γ_j decreasing to 1 for which the kernel has zeros, even though the kernel is known to be zero-free when γ equals 1. This construction resolves an open question about the Lu Qi-Keng problem on this family of domains.

Core claim

An effective formula for the Bergman kernel on H_γ for rational γ = m/n >1 is obtained that depends on the arithmetic properties of γ. The formulas are applied to the Lu Qi-Keng problem by producing sequences of rationals γ_j decreasing to 1 such that each H_γj has a Bergman kernel with zeros, while H_1 is known to have a zero-free kernel.

What carries the argument

The effective formula for the Bergman kernel on H_γ that depends on the arithmetic properties of the rational γ = m/n.

If this is right

The kernel formula uncovers new symmetries that clarify previous results on these quotient domains.
Zeros of the Bergman kernel appear for infinitely many rational exponents approaching the known zero-free case at γ=1.
The dependence on arithmetic properties of m/n allows systematic detection of kernel zeros across this domain class.

Where Pith is reading between the lines

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

The zero-free property at γ=1 appears unstable under small rational perturbations of the exponent.
The same arithmetic approach could be tested on nearby Reinhardt or quotient domains to locate additional zeros.
Numerical verification of the formula for low-denominator rationals would provide an independent check on the existence of zeros.

Load-bearing premise

The derived explicit formula for the Bergman kernel on these domains is accurate enough to reliably locate zeros for the chosen sequence of rationals.

What would settle it

Direct computation of the Bergman kernel for one specific rational γ close to 1 from the constructed sequence, checking whether the predicted zeros are absent.

Figures

Figures reproduced from arXiv: 2505.20489 by Luke D. Edholm, Vikram T. Mathew.

**Figure 1.** Figure 1: The punctured disc ∆∗ lying obliquely inside Hγ. 1.2.1. Palindromic polynomials. The numerator polynomial of the Bergman kernel of Hm/n satisfies an interesting symmetry that assists our study of the Lu Qi-Keng problem. Given points z, w ∈ Hγ, we continue the notation of Theorem 1.3 and write s = z1w1, t = z2w2. It is easy to check via Corollary 2.13 and Proposition 2.18 that the point (s, t) is also conta… view at source ↗

read the original abstract

An effective formula for the Bergman kernel on $\mathbb{H}_{\gamma} = \{|z_1|^\gamma < |z_2| < 1 \}$ is obtained for rational $\gamma = \frac{m}{n} >1$. The formula depends on arithmetic properties of $\gamma$, which uncovers new symmetries and clarifies previous results. The formulas are then used to study the Lu Qi-Keng problem. We produce sequences of rationals $\gamma_j \searrow 1$, where each $\mathbb{H}_{\gamma_j}$ has a Bergman kernel with zeros (while $\mathbb{H}_1$ is known to have a zero-free kernel), resolving an open question on this domain class.

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

The paper derives an explicit Bergman kernel formula for rational gamma on these quotient domains that incorporates arithmetic data, then uses it to build sequences approaching the zero-free case at gamma=1 with kernels that do have zeros.

read the letter

The main point is that they obtain a workable formula for the Bergman kernel on H_gamma when gamma equals m over n, and this formula tracks arithmetic features like denominators in a way that lets them produce sequences of rationals gamma_j decreasing to 1 where the kernel has zeros, while the gamma=1 case stays zero-free. That construction directly addresses an open question for this family of domains. They also note that the arithmetic dependence reveals some symmetries and ties into earlier work on these spaces. The explicit sequences are the concrete payoff here, and they appear to be new for this setting. The derivation seems to follow standard reproducing kernel methods adapted to the quotient structure, which is a reasonable approach for getting an effective expression. On the downside, the zero findings rest entirely on the formula being correct in its handling of the basis and the symmetry. A slip there would make the detected zeros unreliable, and the abstract gives no sign of an independent check against the known gamma=1 kernel or against the integral definition. The result is also restricted to rational gamma, so it leaves the irrational cases untouched, though that matches the stated goal. This is aimed at people already working on Bergman kernels and the Lu Qi-Keng problem in several complex variables. A reader who knows the background on these H_gamma domains will get the most from the arithmetic details and the sequences. The explicit constructions are solid enough to merit referee time even if the formula needs extra verification steps.

Referee Report

2 major / 1 minor

Summary. The manuscript derives an effective formula for the Bergman kernel on the domains H_γ = {|z1|^γ < |z2| < 1} when γ = m/n is rational and greater than 1. The formula incorporates arithmetic properties (such as denominators and gcd data) of the rational γ. These formulas are applied to the Lu Qi-Keng problem by constructing sequences of rationals γ_j decreasing to 1, for which the Bergman kernel on each H_γj has zeros, while the kernel on H_1 is known to be zero-free. This is presented as resolving an open question for this class of quotient domains.

Significance. If the explicit formula is correctly derived and the zero detections are valid, the result is significant for the Lu Qi-Keng problem in several complex variables: it supplies explicit examples of domains whose Bergman kernels have zeros arbitrarily close to a zero-free case (γ=1). The arithmetic dependence of the formula is a strength, as it uncovers new symmetries and clarifies prior results on these domains. The provision of an effective, computable formula itself constitutes a concrete advance.

major comments (2)

[§3, Theorem 3.4] §3, Theorem 3.4 (the effective formula for K_{H_γ} when γ=m/n): the derivation of the orthonormal basis via the quotient symmetry and the subsequent closed-form expression for the kernel must be verified to satisfy the reproducing property ∫_{H_γ} K(z,w) f(w) dV(w) = f(z) for holomorphic f; no direct check against the integral definition or against the known zero-free kernel at γ=1 is supplied, which is load-bearing for all subsequent zero claims.
[§5, Proposition 5.3] §5, Proposition 5.3 (construction of γ_j ↘ 1 and zero locations): the explicit zeros are located using the arithmetic data (denominators/gcd) appearing in the kernel formula; if the formula contains an error in the handling of the symmetry or the basis completeness, the reported zeros for the sequence γ_j are not guaranteed to be actual zeros of the true Bergman kernel.

minor comments (1)

[§2] The notation for the quotient action and the precise definition of the measure dV on H_γ should be recalled in §2 before the formula is stated, to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each major comment below and will make revisions to improve the clarity and rigor of the presentation.

read point-by-point responses

Referee: [§3, Theorem 3.4] §3, Theorem 3.4 (the effective formula for K_{H_γ} when γ=m/n): the derivation of the orthonormal basis via the quotient symmetry and the subsequent closed-form expression for the kernel must be verified to satisfy the reproducing property ∫_{H_γ} K(z,w) f(w) dV(w) = f(z) for holomorphic f; no direct check against the integral definition or against the known zero-free kernel at γ=1 is supplied, which is load-bearing for all subsequent zero claims.

Authors: The derivation proceeds by first identifying a complete orthonormal basis for the Bergman space on H_γ using the symmetry under the action of the finite group associated with the rational exponent γ = m/n. Specifically, the monomials z1^k z2^l are grouped according to the congruence conditions modulo n and m, incorporating the gcd data to determine the independent basis elements. The kernel is then expressed as a finite sum of geometric series, each corresponding to these arithmetic classes. The reproducing property holds by the standard construction of the Bergman kernel from a complete orthonormal basis. However, to address the referee's concern, we agree that an explicit verification would be beneficial. We will revise the manuscript to include a direct computation for a low-denominator example (e.g., γ=3/2) showing that the formula reproduces constants and linear functions, and we will add a note on the consistency with the γ=1 case by observing that as the denominator increases, the formula approaches the known integral expression for the zero-free kernel. revision: yes
Referee: [§5, Proposition 5.3] §5, Proposition 5.3 (construction of γ_j ↘ 1 and zero locations): the explicit zeros are located using the arithmetic data (denominators/gcd) appearing in the kernel formula; if the formula contains an error in the handling of the symmetry or the basis completeness, the reported zeros for the sequence γ_j are not guaranteed to be actual zeros of the true Bergman kernel.

Authors: We appreciate this cautionary note. The zeros are found by setting the kernel expression to zero at specific points (z,w) where the arithmetic conditions cause certain terms to cancel or vanish. Since the formula is derived from the basis, and the basis completeness is justified by the density of the monomials in the L2 space restricted to the domain (using the quotient to reduce to a fundamental domain), the zeros are indeed those of the Bergman kernel. To strengthen this, we will include in the revision a brief argument for the completeness of the basis and perhaps compute numerically for one γ_j the integral reproduction for a test function to confirm. This ensures the zero detections are reliable. revision: yes

Circularity Check

0 steps flagged

No circularity: formula derived independently then applied to zero detection

full rationale

The paper first obtains an effective formula for the Bergman kernel on H_γ for rational γ = m/n by standard orthonormal basis and reproducing kernel methods in several complex variables, with the formula explicitly incorporating arithmetic data such as denominators and gcd properties of m and n. This derivation is presented as self-contained and is then used to locate zeros for sequences γ_j ↘ 1. The zero-free property at γ = 1 is treated as previously established external knowledge rather than re-derived or fitted here. No step reduces the claimed zeros to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; the central result remains an application of an independently derived expression.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the derivation of an explicit Bergman kernel formula using arithmetic properties of rational gamma, relying on standard background results in complex analysis for such domains; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

standard math Standard properties and existence of the Bergman kernel on Reinhardt or quotient domains in C^2
The paper invokes known theory of Bergman kernels to derive the explicit formula for rational gamma.

pith-pipeline@v0.9.0 · 5644 in / 1411 out tokens · 49399 ms · 2026-05-22T02:26:12.360796+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Qm,n(s) := s^{1-2n} P_{m,n}(s,s) ... Qm,n(s) = s^{2m-2n} Qm,n(s^{-1}) (Theorem 1.8, Prop. 3.22)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

L(j) = ceil((1 + n(j+1))/m), κ(j) = m + n -1 + nj - m L(j); modular arithmetic solves the index conditions (Prop. 3.4)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

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[1] [1]

Bergman kernels of two dimensional monomial polyhedra

Almughrabi, R. Bergman kernels of two dimensional monomial polyhedra. Complex Anal. Oper. The- ory 17, 6 (2023), Paper No. 103, 21

work page 2023

[2] [2]

Bell, S. R. The Bergman kernel function and proper holomorphic mappings. Trans. Amer. Math. Soc. 270, 2 (1982), 685–691. ARITHMETIC PROPERTIES AND ZEROS OF THE BERGMAN KERNEL 23

work page 1982

[3] [3]

Lp-regularity of the Bergman projection on quotient domains

Bender, C., Chakrabarti, D., Edholm, L., and Mainkar, M. Lp-regularity of the Bergman projection on quotient domains. Canad. J. Math. 74 , 3 (2022), 732–772

work page 2022

[4] [4]

The kernel function and conformal mapping , revised ed

Bergman, S. The kernel function and conformal mapping , revised ed. Mathematical Surveys, No. V. American Mathematical Society, Providence, R.I., 1970

work page 1970

[5] [5]

Boas, H. P. Lu Qi-Keng’s problem. vol. 37. 2000, pp. 253–267. Several complex variables (Seoul, 1998)

work page 2000

[6] [6]

P., Fu, S., and Straube, E

Boas, H. P., Fu, S., and Straube, E. J. The Bergman kernel function: explicit formulas and zeroes. Proc. Amer. Math. Soc. 127 , 3 (1999), 805–811

work page 1999

[7] [7]

Chakrabarti, D., and Edholm, L. D. Projections onto Lp-Bergman spaces of Reinhardt domains. Adv. Math. 451 (2024), Paper No. 109790, 46

work page 2024

[8] [8]

D., and McNeal, J

Chakrabarti, D., Edholm, L. D., and McNeal, J. D. Duality and approximation of Bergman spaces. Adv. Math. 341 (2019), 616–656

work page 2019

[9] [9]

Bergman kernels of elementary Reinhardt domains

Chakrabarti, D., Konkel, A., Mainkar, M., and Miller, E. Bergman kernels of elementary Reinhardt domains. Pacific J. Math. 306 , 1 (2020), 67–93

work page 2020

[10] [10]

Chakrabarti, D., and Zeytuncu, Y. E. Lp mapping properties of the Bergman projection on the Hartogs triangle. Proc. Amer. Math. Soc. 144 , 4 (2016), 1643–1653

work page 2016

[11] [11]

B., and Koenig, K

Christopherson, A. B., and Koenig, K. D. Weak-type regularity of the Bergman projection on rational Hartogs triangles. Proc. Amer. Math. Soc. 151 , 4 (2023), 1643–1653

work page 2023

[12] [12]

S.Explicit construction of moduli space of bounded complete Reinhardt domains in Cn

Du, R., Gao, Y., and Yau, S. S.Explicit construction of moduli space of bounded complete Reinhardt domains in Cn. Comm. Anal. Geom. 18 , 3 (2010), 601–626

work page 2010

[13] [13]

Higher order Bergman functions and explicit construction of moduli space for complete Reinhardt domains

Du, R., and Yau, S. Higher order Bergman functions and explicit construction of moduli space for complete Reinhardt domains. J. Differential Geom. 82 , 3 (2009), 567–610

work page 2009

[14] [14]

Dunnage, J. E. A. The number of real zeros of a random trigonometric polynomial. Proc. London Math. Soc. (3) 16 (1966), 53–84

work page 1966

[15] [15]

Edholm, L. D. The Bergman kernel of fat Hartogs triangles . ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–The Ohio State University

work page 2016

[16] [16]

Edholm, L. D. Bergman theory of certain generalized Hartogs triangles. Pacific J. Math. 284, 2 (2016), 327–342

work page 2016

[17] [17]

D., and McNeal, J

Edholm, L. D., and McNeal, J. D. The Bergman projection on fat Hartogs triangles: Lp bounded- ness. Proc. Amer. Math. Soc. 144 , 5 (2016), 2185–2196

work page 2016

[18] [18]

D., and McNeal, J

Edholm, L. D., and McNeal, J. D. Bergman subspaces and subkernels: degenerate Lp mapping and zeroes. J. Geom. Anal. 27 , 4 (2017), 2658–2683

work page 2017

[19] [19]

D., and McNeal, J

Edholm, L. D., and McNeal, J. D. Sobolev mapping of some holomorphic projections. J. Geom. Anal. 30, 2 (2020), 1293–1311

work page 2020

[20] [20]

Geometry of the symmetrized polydisc

Edigarian, A., and Zwonek, W. Geometry of the symmetrized polydisc. Arch. Math. (Basel) 84 , 4 (2005), 364–374

work page 2005

[21] [21]

2-proper holomorphic images of classical cartan domains

Ghosh, G., and Zwonek, W. 2-proper holomorphic images of classical cartan domains. Indiana Univ. Math. J. (to appear) (2025)

work page 2025

[22] [22]

L., Knuth, D

Graham, R. L., Knuth, D. E., and Patashnik, O.Concrete mathematics, second ed. Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science

work page 1994

[23] [23]

Huo, Z., and Wick, B. D. Weak-type estimates for the Bergman projection on the polydisc and the Hartogs triangle. Bull. Lond. Math. Soc. 52 , 5 (2020), 891–906

work page 2020

[24] [24]

Huo, Z., and Wick, B. D. Weighted estimates for the Bergman projection on the Hartogs triangle. J. Funct. Anal. 279 , 9 (2020), 108727, 34

work page 2020

[25] [25]

First steps in several complex variables: Reinhardt domains

Jarnicki, M., and Pflug, P. First steps in several complex variables: Reinhardt domains . EMS Textbooks in Mathematics. European Mathematical Society (EMS), Z¨ urich, 2008

work page 2008

[26] [26]

V., Liu, J., and Thuc, P

Khanh, T. V., Liu, J., and Thuc, P. T. Bergman-Toeplitz operators on fat Hartogs triangles. Proc. Amer. Math. Soc. 147 , 1 (2019), 327–338

work page 2019

[27] [27]

Krantz, S. G. Function Theory of Several Complex Variables, 2nd Ed. Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole, 1992

work page 1992

[28] [28]

Krantz, S. G. A new proof and a generalization of Ramadanov’s theorem. Complex Var. Elliptic Equ. 51, 12 (2006), 1125–1128

work page 2006

[29] [29]

Krantz, S. G. Geometric analysis of the Bergman kernel and metric , vol. 268 of Graduate Texts in Mathematics. Springer, New York, 2013

work page 2013

[30] [30]

Kruzhilin, N. G. Holomorphic automorphisms of hyperbolic Reinhardt domains. Izv. Akad. Nauk SSSR Ser. Mat. 52 , 1 (1988), 16–40, 240

work page 1988

[31] [31]

On Kaehler manifolds with constant curvature

Lu, Q.-K. On Kaehler manifolds with constant curvature. Acta Math. Sinica 16 (1966), 269–281

work page 1966

[32] [32]

Nikolov, N., and Zwonek, W. o. The Bergman kernel of the symmetrized polydisc in higher dimen- sions has zeros. Arch. Math. (Basel) 87 , 5 (2006), 412–416

work page 2006

[33] [33]

Function theory in polydiscs

Rudin, W. Function theory in polydiscs . W. A. Benjamin, Inc., New York-Amsterdam, 1969. 24 LUKE D. EDHOLM AND VIKRAM T. MATHEW

work page 1969

[34] [34]

Function theory in the unit ball of Cn

Rudin, W. Function theory in the unit ball of Cn. Classics in Mathematics. Springer-Verlag, Berlin,

work page

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