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arxiv: 2606.24833 · v1 · pith:NTEGRNMVnew · submitted 2026-06-23 · 🧮 math.DS

Biggest bounded type Siegel disks of monic polynomials include those that stick to all critical points

Xavier Buff , Arnaud Ch\'eritat , Pascale Roesch This is my paper

Pith reviewed 2026-06-25 21:43 UTC · model grok-4.3

classification 🧮 math.DS
keywords Siegel disksmonic polynomialsbounded type irrationalconformal radiuscritical pointsDouady conjecturerotation number
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The pith

Among monic polynomials with a period-1 Siegel disk of bounded type rotation number, the maximum conformal radius is achieved when all critical points lie on the boundary.

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

The paper proves that for every degree d at least 2 and every bounded type irrational rotation number θ, in the space of monic polynomials with a fixed Siegel disk of rotation number θ, the polynomials that maximize the conformal radius of the disk have all critical points on its boundary. This holds because the maximum locus in that space includes such polynomials. The result is then applied to simplify Douady's conjecture on the Bruno condition by reducing it to a weaker statement.

Core claim

For all degree d≥2 and all bounded type irrational θ, in the space of monic polynomials having a period 1 Siegel disk Δ of rotation number θ, the maximum locus of the conformal radius of Δ with respect to its fixed point contains polynomials having all critical points on the boundary of Δ.

What carries the argument

The maximum locus of the conformal radius of the Siegel disk with respect to its fixed point, within the space of monic polynomials of given degree and rotation number.

If this is right

  • The result applies directly to reduce Douady's conjecture on the optimality of the Bruno condition to a weaker statement.
  • Such maximizing polynomials exist for every degree d≥2 and bounded type θ.
  • The conformal radius is maximized at configurations where critical points stick to the boundary.

Where Pith is reading between the lines

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

  • If true, this suggests that the 'biggest' Siegel disks are those fully 'stuck' to their critical points, potentially simplifying computations of maximal sizes.
  • Extensions might include checking whether this holds for non-monic polynomials or higher period cycles.
  • Connections to the dynamics on the boundary could be explored to see how critical points influence the rotation.

Load-bearing premise

The space of monic polynomials of degree d with a period-1 Siegel disk of rotation number θ is non-empty and the conformal radius attains its supremum on a non-empty locus within that space.

What would settle it

A counterexample would be a monic polynomial with a bounded type period-1 Siegel disk whose conformal radius exceeds that of all polynomials with critical points on the boundary.

Figures

Figures reproduced from arXiv: 2606.24833 by Arnaud Ch\'eritat, Pascale Roesch, Xavier Buff.

Figure 1
Figure 1. Figure 1: See the caption in Section 5 consists in a finite union over pairs of indices i < j of critical points, of d − 2- dimensional topological submanifolds foliated by a one real parameter family of d − 3-dimensional complex submanifolds, the parameter being the relative angle between the two critical points on ∂∆(P). And T is the sum over i, j of the integral with respect to the Lebesgue measure on the relativ… view at source ↗
Figure 2
Figure 2. Figure 2: See the caption in Section 5 [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
read the original abstract

We prove that for all degree $d\geq 2$ and all bounded type irrational $\theta$, in the space of monic polynomials having a period $1$ Siegel disk $\Delta$ of rotation number $\theta$, the maximum locus of the conformal radius of $\Delta$ with respect to its fixed point contains polynomials having all critical points on the boundary of $\Delta$. We apply this to reduce a conjecture of Douady (optimality of the Bruno condition) to a weaker statement.

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 / 2 minor

Summary. The manuscript proves that for every degree d≥2 and every bounded-type irrational rotation number θ, in the parameter space of monic polynomials possessing a period-1 Siegel disk Δ of rotation number θ, the locus of polynomials that maximize the conformal radius of Δ (measured at the fixed point) contains polynomials for which every critical point lies on ∂Δ. The result is then used to reduce Douady’s conjecture on the optimality of the Brjuno condition to a weaker auxiliary statement.

Significance. If correct, the characterization supplies a concrete geometric description of extremal Siegel disks within each fixed-rotation-number slice of the polynomial parameter space. The reduction of Douady’s conjecture isolates a potentially more tractable sub-problem and therefore constitutes a genuine technical advance, provided the attainment of the supremum is rigorously established. The argument is presented as direct and non-circular.

major comments (2)
  1. [Introduction / statement of Theorem 1] The central claim presupposes that the conformal radius, viewed as a function on the space of monic degree-d polynomials with a fixed period-1 Siegel disk of rotation number θ, attains its supremum. No compactness, properness, or coercivity argument establishing attainment is visible in the abstract or in the statement of the main theorem; if the supremum is not attained the maximum locus is empty and the asserted containment is vacuous. This issue is load-bearing for both the main theorem and the subsequent reduction of Douady’s conjecture.
  2. [§2 (parameter space definition)] The non-emptiness of the space of monic polynomials of degree d with a period-1 Siegel disk of rotation number θ is tacitly assumed when the maximum locus is discussed. While the space is known to be non-empty for certain θ, the manuscript does not supply a reference or self-contained argument that the space remains non-empty for every bounded-type θ under consideration.
minor comments (2)
  1. [§1] Notation for the conformal radius r(Δ) and its dependence on the fixed point should be made uniform throughout; at present the same symbol is used both for the radius at the fixed point and for the radius relative to other base points.
  2. [§4] The reduction step to Douady’s conjecture would benefit from an explicit statement of the weaker auxiliary claim that remains to be proved.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Introduction / statement of Theorem 1] The central claim presupposes that the conformal radius, viewed as a function on the space of monic degree-d polynomials with a fixed period-1 Siegel disk of rotation number θ, attains its supremum. No compactness, properness, or coercivity argument establishing attainment is visible in the abstract or in the statement of the main theorem; if the supremum is not attained the maximum locus is empty and the asserted containment is vacuous. This issue is load-bearing for both the main theorem and the subsequent reduction of Douady’s conjecture.

    Authors: We agree that the manuscript does not contain an explicit compactness or coercivity argument establishing that the supremum of the conformal radius is attained. This is a substantive omission. In the revised version we will insert a short subsection (or appendix) proving attainment: the relevant slice of coefficient space is closed, the conformal radius tends to zero as ||P|| → ∞ or as critical points escape to infinity, and sublevel sets are therefore compact. This makes the maximum locus non-empty and the containment statement non-vacuous. The added argument will be independent of the later reduction of Douady’s conjecture. revision: yes

  2. Referee: [§2 (parameter space definition)] The non-emptiness of the space of monic polynomials of degree d with a period-1 Siegel disk of rotation number θ is tacitly assumed when the maximum locus is discussed. While the space is known to be non-empty for certain θ, the manuscript does not supply a reference or self-contained argument that the space remains non-empty for every bounded-type θ under consideration.

    Authors: The referee is correct that non-emptiness is assumed without citation. For every irrational θ the existence of at least one monic polynomial of degree d possessing a period-1 Siegel disk of rotation number θ is a standard result (obtained, for example, by quasiconformal surgery from the quadratic case or by direct construction). In the revision we will add an explicit reference to this existence theorem, restricted if necessary to the bounded-type case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct proof reduces external conjecture

full rationale

The abstract states a direct proof that, in the space of monic polynomials with a period-1 Siegel disk of bounded-type irrational rotation number θ, the maximum locus of the conformal radius contains polynomials with all critical points on ∂Δ, and applies this to reduce Douady’s conjecture to a weaker statement. No quoted equations or steps exhibit self-definitional reduction, fitted parameters renamed as predictions, load-bearing self-citations, or ansatz smuggling. The implicit prerequisites (non-emptiness of the space and attainment of the supremum) are logical prerequisites for the maximum locus to be non-empty, not circular reductions by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard existence theory for Siegel disks of bounded-type rotation numbers and on the well-definedness of the conformal radius in the space of monic polynomials; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Existence and basic properties of period-1 Siegel disks for bounded-type irrational rotation numbers in the space of monic polynomials
    The statement presupposes that the relevant space is non-empty and that the conformal radius is a well-defined continuous function on it.

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Reference graph

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