Brennan Conjecture for Basin of Attraction at Infinity

Yigang Zheng

arxiv: 2604.12240 · v1 · submitted 2026-04-14 · 🧮 math.DS

Brennan Conjecture for Basin of Attraction at Infinity

Yigang Zheng This is my paper

Pith reviewed 2026-05-10 16:05 UTC · model grok-4.3

classification 🧮 math.DS

keywords Brennan conjecturebasin of attractionmonic polynomialcomplex dynamicspolynomial iterationunbounded domains

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The pith

Brennan conjecture holds for basins of attraction of monic polynomials with sufficiently small non-leading coefficients.

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 Brennan conjecture is true for a specific family of domains in the complex plane. These domains are the basins of attraction at infinity for monic polynomials of any degree m, provided the remaining coefficients are small enough. The result enlarges the class of domains where the conjecture is known to hold by building on an earlier theorem for a narrower set of polynomials. Readers interested in complex dynamics or potential theory would care because it supplies an explicit, infinite collection of concrete domains that satisfy a classical integrability statement.

Core claim

We prove that for any monic polynomial of degree m with sufficiently small non-leading coefficients, Brennan Conjecture holds for its basin of attraction at infinity. This extends the result of Baranski, Volberg, and Zdunik to a broader class of polynomials.

What carries the argument

The basin of attraction at infinity for the monic polynomial, used as the domain on which the Brennan conjecture is verified.

If this is right

The conjecture is verified for all monic polynomials whose non-leading coefficients lie below a degree-dependent threshold.
The result applies uniformly across every degree m once the coefficients are sufficiently small.
It enlarges the known collection of domains satisfying the conjecture beyond the cases treated by Baranski, Volberg, and Zdunik.

Where Pith is reading between the lines

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

The small-coefficient condition might be removable if a uniform bound independent of m can be found.
Numerical checks on low-degree examples with tiny constant terms could test how sharp the smallness requirement is.
Similar arguments may apply to other conjectures about integrals over unbounded dynamical domains.

Load-bearing premise

The non-leading coefficients of the monic polynomial must be small enough, with the required bound possibly depending on the degree.

What would settle it

A monic polynomial of some degree m whose non-leading coefficients are small enough yet whose basin of attraction at infinity violates the Brennan conjecture would serve as a direct counterexample.

read the original abstract

This paper investigates the Brennan Conjecture for domains $\Omega$ that arise as basins of attraction of a polynomial. We extend the result of Baranski, Volberg, and Zdunik to a broader class of polynomials. We prove that for any monic polynomial of degree $m$ with sufficiently small non-leading coefficients, Brennan Conjecture holds for its basin of attraction.

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 extends the Brennan conjecture to basins of monic polynomials with small lower coefficients but leaves the smallness threshold unquantified, making the result non-effective.

read the letter

The central claim is that Brennan's conjecture holds for the basin of attraction at infinity of any monic polynomial of degree m when the non-leading coefficients are sufficiently small. This extends the Baranski-Volberg-Zdunik result from the pure monomial z^m to a neighborhood in coefficient space, using the known case as the base for a perturbation argument on the Riemann map and its derivative integrability. The choice of domain ties the conjecture to polynomial dynamics in a direct way, which is a reasonable next step within the existing literature on conformal mappings of these basins. The work stays grounded in prior results on the integrability of |φ'|^(-p) and does not introduce new frameworks or first-principles derivations. That keeps the advance modest but legitimate for the specific family considered. The main limitation is the non-quantified threshold. Without an explicit δ(m) that can be computed or bounded from the proof, the statement cannot be checked for any concrete polynomial, and the result stays theoretical rather than usable. The argument likely proceeds by continuity or compactness around the monomial case without effective rates, which matches the stress-test concern exactly. The basin restriction itself adds no extra quantitative control to fix this. The citations follow the standard references on the conjecture and show no circularity. This paper is for specialists already working on the Brennan conjecture or on integrability questions in holomorphic dynamics. A reader seeking broad techniques or effective constants will find the scope narrow, but the claim is precise enough to be worth checking. It deserves peer review because the extension is new within its literature and the proof steps can be examined for gaps in the estimates. I would send it to referees with a request to address the effectivity of the smallness condition.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove that Brennan's conjecture holds for the basin of attraction at infinity of any monic polynomial p(z) = z^m + lower-degree terms, provided the non-leading coefficients are sufficiently small. This extends the known case for the monomial z^m (due to Baranski, Volberg, and Zdunik) via a perturbative argument around that limiting case.

Significance. If the central claim holds with an effective threshold, the result would establish local stability of the Brennan integrability condition under small perturbations of the monomial in coefficient space. This is a meaningful extension because the exterior of the unit disk satisfies the conjecture by direct computation, and a neighborhood result would cover a positive-measure set of polynomials whose Julia sets are close to the circle.

major comments (2)

[Abstract and §1] Abstract and §1 (Introduction): the statement that the result holds 'for sufficiently small non-leading coefficients' does not supply an explicit δ(m) > 0 or a computable procedure to obtain one. The proof sketch (compactness or limiting argument around z^m) therefore leaves the claim non-effective; for any concrete polynomial it is impossible to verify whether the hypothesis is satisfied without an a-priori bound.
[§3] §3 (Main argument): the handling of the 'sufficiently small' regime must be checked for uniformity in m. If the error estimates or the radius of the neighborhood in coefficient space depend on m in a non-explicit way (e.g., via compactness without quantitative control), the result reduces to an existence statement rather than a verifiable theorem.

minor comments (2)

[§2] Notation for the basin Ω and the Riemann map φ should be introduced with a precise definition of the domain of integration (exterior of the filled Julia set) before the integrability statement is written.
[References] The reference list should include the precise citation for Baranski–Volberg–Zdunik and any subsequent works on Brennan's conjecture for polynomial basins.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the non-effective character of the main result. We agree that the current proof yields only an existential statement and will revise the paper accordingly to make this explicit.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1 (Introduction): the statement that the result holds 'for sufficiently small non-leading coefficients' does not supply an explicit δ(m) > 0 or a computable procedure to obtain one. The proof sketch (compactness or limiting argument around z^m) therefore leaves the claim non-effective; for any concrete polynomial it is impossible to verify whether the hypothesis is satisfied without an a-priori bound.

Authors: We concur that the argument is non-effective. The proof relies on a limiting procedure as the non-leading coefficients approach zero, combined with the known Brennan conjecture for the monomial z^m, and invokes compactness in the coefficient space without quantitative control. Consequently, no explicit δ(m) or algorithm for determining it is obtained. We will revise the abstract and §1 to state clearly that the theorem asserts the existence of some δ(m) > 0 (depending on m) such that the conjecture holds whenever the non-leading coefficients are smaller than δ(m) in modulus, while emphasizing that the result is not constructive. revision: yes
Referee: [§3] §3 (Main argument): the handling of the 'sufficiently small' regime must be checked for uniformity in m. If the error estimates or the radius of the neighborhood in coefficient space depend on m in a non-explicit way (e.g., via compactness without quantitative control), the result reduces to an existence statement rather than a verifiable theorem.

Authors: The referee correctly identifies that the neighborhood radius and error estimates in §3 depend on m through the degree and the underlying compactness or continuity arguments, without explicit quantitative bounds. No uniformity in m is claimed or obtained. We will insert a clarifying paragraph in §3 noting that, for each fixed m, a sufficiently small neighborhood exists, but its size is determined non-constructively and may deteriorate with m. This preserves the result as an existence theorem for each degree. revision: yes

standing simulated objections not resolved

We cannot supply an explicit or computable value of δ(m) > 0, as this would require a fully quantitative version of the perturbation argument that is not provided by the compactness-based approach in the manuscript.

Circularity Check

0 steps flagged

No circularity detected; extension of external prior result

full rationale

The provided abstract and context describe an extension of the Baranski-Volberg-Zdunik result to monic polynomials whose non-leading coefficients lie in a sufficiently small ball. No self-citations appear, no parameters are fitted to data and then relabeled as predictions, and no ansatz or uniqueness claim is imported from the authors' own prior work. The 'sufficiently small' hypothesis is an existence statement typical of perturbation arguments around the monomial z^m (whose basin satisfies Brennan's conjecture by direct computation); it does not reduce the target integrability statement to a tautology or to a fitted input. The derivation is therefore independent of its own outputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The proof presumably relies on standard results from complex analysis and conformal mapping, but these cannot be audited.

pith-pipeline@v0.9.0 · 5335 in / 1037 out tokens · 45849 ms · 2026-05-10T16:05:32.343691+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Brennan’s conjecture and the Mandelbrot set

Krzysztof Bara´ nski, Alexander Volberg, and Anna Zdunik. “Brennan’s conjecture and the Mandelbrot set”. In:International Mathematics Research Notices1998.12 (Jan. 1998), pp. 589–600.issn: 1073-7928.doi:10 . 1155 / S1073792898000373. eprint:https://academic.oup.com/imrn/article-pdf/1998/12/589/2600721/ 1998-12-589.pdf.url:https://doi.org/10.1155/S1073792898000373

work page doi:10.1155/s1073792898000373 1998
[2]

On Brennan

D. Bertilsson. “On Brennan”s conjecture in conformal mapping”. In:mathematics (1999)

work page 1999
[3]

The Integrability of the Derivative in Conformal Mapping

James E Brennan. “The Integrability of the Derivative in Conformal Mapping”. In: Journal of the London Mathematical Society2 (1978), pp. 261–272

work page 1978
[4]

Complex dynamics / Lennart Car- leson, Theodore W. Gamelin

Lennart Carleson and Theodore W. Gamelin. “Complex dynamics / Lennart Car- leson, Theodore W. Gamelin”. In: ()

work page
[5]

Weighted Bergman spaces and the in- tegral means spectrum of conformal mappings

H˚ akan Hedenmalm and Serguei Shimorin. “Weighted Bergman spaces and the in- tegral means spectrum of conformal mappings”. In:Duke Math. J.127.2 (2005), pp. 341–393.issn: 0012-7094,1547-7398.doi:10.1215/S0012-7094-04-12725-3. url:https://doi.org/10.1215/S0012-7094-04-12725-3

work page doi:10.1215/s0012-7094-04-12725-3 2005
[6]

Koebe distortion theorem

M. Murugan. “Koebe distortion theorem”. In: (2019). 12

work page 2019
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On polynomial approximation inA q(D)

Thomas, A., and Metzger. “On polynomial approximation inA q(D)”. In:Proceed- ings of the American Mathematical Society37.2 (1973), pp. 468–470. 13

work page 1973

[1] [1]

Brennan’s conjecture and the Mandelbrot set

Krzysztof Bara´ nski, Alexander Volberg, and Anna Zdunik. “Brennan’s conjecture and the Mandelbrot set”. In:International Mathematics Research Notices1998.12 (Jan. 1998), pp. 589–600.issn: 1073-7928.doi:10 . 1155 / S1073792898000373. eprint:https://academic.oup.com/imrn/article-pdf/1998/12/589/2600721/ 1998-12-589.pdf.url:https://doi.org/10.1155/S1073792898000373

work page doi:10.1155/s1073792898000373 1998

[2] [2]

On Brennan

D. Bertilsson. “On Brennan”s conjecture in conformal mapping”. In:mathematics (1999)

work page 1999

[3] [3]

The Integrability of the Derivative in Conformal Mapping

James E Brennan. “The Integrability of the Derivative in Conformal Mapping”. In: Journal of the London Mathematical Society2 (1978), pp. 261–272

work page 1978

[4] [4]

Complex dynamics / Lennart Car- leson, Theodore W. Gamelin

Lennart Carleson and Theodore W. Gamelin. “Complex dynamics / Lennart Car- leson, Theodore W. Gamelin”. In: ()

work page

[5] [5]

Weighted Bergman spaces and the in- tegral means spectrum of conformal mappings

H˚ akan Hedenmalm and Serguei Shimorin. “Weighted Bergman spaces and the in- tegral means spectrum of conformal mappings”. In:Duke Math. J.127.2 (2005), pp. 341–393.issn: 0012-7094,1547-7398.doi:10.1215/S0012-7094-04-12725-3. url:https://doi.org/10.1215/S0012-7094-04-12725-3

work page doi:10.1215/s0012-7094-04-12725-3 2005

[6] [6]

Koebe distortion theorem

M. Murugan. “Koebe distortion theorem”. In: (2019). 12

work page 2019

[7] [7]

On polynomial approximation inA q(D)

Thomas, A., and Metzger. “On polynomial approximation inA q(D)”. In:Proceed- ings of the American Mathematical Society37.2 (1973), pp. 468–470. 13

work page 1973