Univalent Polynomials and Hubbard Trees

Kirill Lazebnik; Nikolai G. Makarov; Sabyasachi Mukherjee

REVIEW 2 major objections 5 minor 31 references

Univalent Polynomials and Hubbard Trees

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A single finite bi-angled tree uniquely encodes each extremal univalent polynomial, and the same trees classify all critically fixed anti-polynomials.

desk verdict Strong and likely correct classification, but the uniqueness half of Theorem A has a load-bearing gap around the combined quasiconformal map Ψ0. read the letter →

arxiv 1908.05813 v2 pith:WKIWJTEY submitted 2019-08-16 math.CV math.DS

Kirill Lazebnik , Nikolai G. Makarov , Sabyasachi Mukherjee This is my paper · ORCID

classification math.CVmath.DS MSC 30C5537F1030C62

keywords univalentpolynomialsquadraturedomainsSchwarzreflectionmapsbi-angledtreesHubbardanti-holomorphiccriticallyfixedpointspinching

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies rational maps $f$ of degree $d+1$ that are univalent in the exterior of the unit disk and whose image of the unit circle has the maximal possible number of cusps ($d+1$) and double points ($d-2$). Its main theorem is a canonical bijection between such extremal maps up to rotation, finite plane trees called bi-angled trees with $d-1$ vertices, and anti-holomorphic polynomials of degree $d$ whose $d-1$ critical points are distinct and fixed. The bi-angled tree is a tree whose edges meet at $2\pi/3$ or $4\pi/3$; it records exactly how the $d-1$ interior components of the curve's complement touch. If the theorem is right, every extremal unbounded quadrature domain is encoded by a tiny finite tree, and the same trees give a complete classification of critically fixed anti-polynomials, connecting harmonic-polynomial sharpness to holomorphic dynamics.

What carries the argument

The central combinatorial object is the bi-angled tree: a tree with vertices of degree at most three, embedded so that all edges are straight segments meeting at angles $2\pi/3$ or $4\pi/3$, with an angle function encoding the cyclic order around each vertex. The load-bearing geometric construction is the pinching theorem: starting from the base map $f_0(z)=z-\frac{1}{d}z^d$ whose image is a hypocycloid, the paper uses Schwarz-reflection-invariant Beltrami coefficients, supported on preimages of a quadrilateral under the Schwarz reflection map, to stretch selected arcs and force specified pairs of arcs to meet, creating double points one at a time while preserving all other incidences. An augmented tree, obtained by inserting blue vertices into the bi-angled tree, lists which cusps and double points lie on which fundamental tiles and hence which arcs must be pinched. For uniqueness, the machinery is a pullback argument: a conformal map between two droplets with isomorphic trees is shown to be asymptotically linear at the singular points, extended to a global quasiconformal map, then lifted by iterates of the Schwarz reflections; the limit is conformal off a zero-area set and hence affine.

What would settle it

Take two extremal quadrature domains with isomorphic bi-angled trees and compute, in the coordinates that straighten each Schwarz reflection near infinity, the landing point of the fixed external ray corresponding to a shared cusp; if the two coordinates place that landing point at images that are not mapped to each other by the droplet homeomorphism, then no map with the required combined agreement can exist, and the uniqueness proof would need a different extension.

Watch

Extended reading notes

Core claim

Every extremal $f\in\Sigma^*_d$—a rational map of degree $d+1$, univalent outside the closed unit disk, with $d+1$ cusps and $d-2$ double points on $f(\mathbb T)$—determines a droplet $\widehat{\mathbb C}\setminus f(\widehat{\mathbb C}\setminus \mathbb D)$, whose $d-1$ interior components are topological triangles. Treating each triangle as a vertex and joining two vertices when the triangles touch creates a bi-angled tree $\mathcal T(f)$. The paper proves two directions: first, every abstract bi-angled tree with $d-1$ vertices is realized by such an $f$ (surjectivity, via a pinching procedure), and second, two such $f$ realize isomorphic trees only if they differ by multiplication by a $(d+1)$-st root of unity (injectivity, via a quasiconformal pullback argument). Thus affine equivalence classes of extremal unbounded quadrature domains are in bijection with isomorphism classes of bi-angled trees. The same trees arise as the angled Hubbard trees of anti-holomorphic polynomials with $d-1$ distinct fixed critical points, and the paper invokes a realization theorem for angled trees to obtain the bijection with affine conjugacy classes of these anti-polynomials.

Load-bearing premise

Toward proving uniqueness, the argument needs a global quasiconformal map that simultaneously matches the prescribed homeomorphism between the two droplets and matches the coordinate straightening of the two Schwarz reflections near infinity and along the chosen fixed ray; the paper asserts such a combined extension exists, but the lemma it cites proves only that the droplet homeomorphism admits a quasiconformal extension.

Editorial extensions

If this is right

Every bi-angled tree with $d-1$ vertices occurs in every degree; extremal Suffridge polynomials therefore exist in abundance rather than only by a convexity argument.
Two extremal unbounded quadrature domains are affinely equivalent exactly when their bi-angled trees are isomorphic, so the geometric class is completely classified by finite trees.
The classification extends to critically fixed anti-polynomials: affine conjugacy classes of degree-$d$ anti-polynomials with $d-1$ distinct fixed critical points are in bijection with bi-angled trees with $d-1$ vertices.
In the bounded class $S^*_d$, the same methods give that extremal polynomials correspond to rooted binary trees with $d-2$ vertices, and the count is the Catalan number $\frac{1}{d-1}\binom{2d-4}{d-2}$.
The shared tree invariant supplies a direct route from an extremal quadrature domain to a critically fixed anti-polynomial, making the dynamics of the Schwarz reflection map combinatorially accessible from the tree.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The classification implies a count of affine equivalence classes of extremal unbounded quadrature domains: it is the number of bi-angled trees with $d-1$ vertices, which could be enumerated by a simple recurrence on plane trees.
Because the proof realizes trees by pinching arcs, a natural testable extension is that non-extremal limits—where fewer double points are created—should be parameterized by the same trees with some edges left unpinched, i.e., bi-angled forests with marked active edges.
The bijection between quadrature domains and critically fixed anti-polynomials suggests an explicit dictionary: properties of the Schwarz reflection tiling, such as external ray landing patterns, should be readable from the Hubbard tree's internal-ray angles, and conversely.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the takeaway: this is a genuinely substantial paper. The bi-angled tree parameterization of Suffridge polynomials and the bijection with CS anti-polynomials is a new result that opens a clean bridge between quadrature domains and anti-holomorphic dynamics. The counting corollary for S*_d is a nice bonus. The pinching technique in Section 4 is the real novelty; it is more elementary than the Thurston-based existence proofs, and the geometric Lemmas 5.2–5.3 are carefully done.

The soft spot that matters is in the proof of Theorem 5.1. The rigidity argument needs a quasiconformal map Ψ0 that both extends the tile-map Ψ and coincides with the Böttcher conjugacy on a neighborhood of infinity and on the fixed ray. The text says Lemma 5.4 guarantees this. It does not. Lemma 5.4 only proves a quasiconformal extension of Ψ; it says nothing about matching the external dynamics. Since the subsequent pullback construction uses the conjugacy near infinity to pin down the lifts and force the ray normalization, this is a load-bearing gap, not a cosmetic detail. I think it is repairable—one would need to construct a genuine interpolation, perhaps using the asymptotic linearity from Lemma 5.3 to glue the two maps across a collar—but as written it is an unsupported assertion.

Two smaller issues, in proportion. First, the recursive pinching in Section 4 is only fully written for the first two steps; the iteration to |S| pinchings is hand-waved as 'completely analogous.' That is likely fine for the intended reader, but it is still a sketch in a central proof. Second, Section 7 on the S*_d side is explicitly an outline, with Theorem 7.8 just an 'adaptation' of Theorem 5.1. The Catalan count is a direct consequence, so the result stands or falls with the rigidity statement.

I don't think any of this makes the main classification false. The architecture of the proof is sound, and the use of Poirier's Hubbard tree classification on the anti-polynomial side is appropriate. The heavy reliance on the authors' earlier papers is not circular; those results are cited by name and are independent.

Who is this for? People working in geometric function theory, quadrature domains, and complex dynamics. It deserves a serious referee. My recommendation: send it out, but ask the authors to fix the Ψ0 gap—either by proving a proper interpolation lemma or by restructuring the rigidity argument—and to expand the sketchy induction in Section 4. After those revisions, I would expect the paper to be accepted.

Referee Report

2 major / 5 minor

Summary. The paper studies external polynomials in Σ*_d whose image of the unit circle has the maximal number d−2 of double points (Suﬀridge polynomials), and proves a canonical bijection between such polynomials modulo the Z_{d+1}-action, isomorphism classes of bi-angled trees with d−1 vertices, and affine conjugacy classes of degree-d anti-polynomials with d−1 distinct fixed critical points (Crofoot–Sarason anti-polynomials). The proof combines a quasiconformal pinching construction for existence (Theorem 4.1), a pullback/quasiconformal rigidity argument for uniqueness (Theorem 5.1), and Poirier's classification of angled Hubbard trees for the anti-polynomial side. A parallel bijection for the class S*_d is used to enumerate extremal functions by Catalan numbers (Theorem B).

Significance. If the classification is correct, it is a substantial result: it completely describes all extremal unbounded quadrature domains in combinatorial terms and establishes a new bridge between quadrature domains and anti-holomorphic dynamics. The pinching technique for Σ*_d is a genuine methodological novelty, and Theorem B gives an elegant Catalan count that is likely to be of independent interest. The paper is also careful in several places: cusp types and tangency orders are proved from scratch (Propositions 2.9 and 2.10), the quasiconformal closure of the family is stated precisely (Proposition 3.3), and the use of Poirier's realization theorem is explicit. The main obstruction to accepting the paper as it stands is the incomplete proof of the combined quasiconformal extension in Theorem 5.1, which is load-bearing for the uniqueness half of the central classification.

major comments (2)

[§5.1, proof of Theorem 5.1 (first paragraph after Lemma 5.4)] The proof defines a K-quasiconformal map Ψ0 that agrees with Ψ on the droplet ~T and with τ∘τ~^{-1} on U∪R0(~σ), and states: 'The existence of such a map is guaranteed by Lemma 5.4.' Lemma 5.4, however, only establishes a quasiconformal extension of Ψ to the sphere; it says nothing about matching the Böttcher conjugacy on the invariant neighborhood U or along the fixed ray R0(~σ), and no interpolation across the region between U and the droplet is constructed. This is not a cosmetic omission: the subsequent lifts Ψ1,Ψ2,... and the normalization Ψ1(~ζ0)=ζ0 depend on having a single K-quasiconformal map that simultaneously realizes both prescribed behaviors. Without such a map, the identity (32), the statement Ψ1=Ψ0 on U, and the conclusion that the limiting Ψ∞ is conformal (hence affine) are unsupported. A separate gluing/interpolation lemma is needed that starts from Lemma 5.4 and the asymptotic linearity of Lemma 5.3 and produces Ψ0 with the stated boundary data and uniformly bounded dilatation; the manuscript should also specify how large U is and how the fixed ray is treated in the interpolation.
[§4.2, proof of Theorem 4.1 (after Proposition 4.13)] The recursive pinching construction is not written for a general bi-angled tree. After the second step, the proof says 'subsequent steps are completely analogous' and stops. Since the construction is recursive over the set S_T of pinched pairs, one needs an induction statement that after each pinching step (i) the previously created double points persist in the limit, (ii) no new intersections are created outside the prescribed pairs, and (iii) the quadrilateral chosen for the next pair is admissible. The second step already requires a separate treatment when one of the chosen arcs has an endpoint at the existing double point; later steps may involve several pre-existing double points, and the curvature/modulus arguments in Proposition 4.13 would need to be re-run in that generality. This is a completeness gap in the existence half of Theorem A. The method is plausible and the gap appears repairable, but as written the proof relies on an unstated induction.

minor comments (5)

[§5.1, Böttcher coordinate paragraph] The text says the conformal isomorphism τ conjugates z^d to σ, but σ is anti-meromorphic; the standard statement is a conjugacy to \bar z^d (or, after passing to σ^2, to z^{d^2}). Please correct the statement or the convention.
[Lemma 5.4, proof] The sentence 'in fact, we have showed that \hatΨ : T→T is C1' is stronger than what is proved: Lemma 5.3 gives asymptotic linearity at the singular points and conformality away from them, which implies local quasisymmetry but not global C1 regularity. Please rephrase.
[Lemma 5.4 and Section 5 generally] The symbol T is used both for the droplet \hat C\\Ω (Definition 2.15) and for the unit circle in the lifting diagram in Lemma 5.4. This notational clash makes the proof harder to follow; use \mathbb T or a different letter for the unit circle.
[Definition 2.19] The phrase 'allowing for edges to decrease in length so as to avoid self-intersection' describes the recursive construction of the bi-angled tree informally. Since the bi-angled tree is a central invariant, a formal statement that this procedure terminates and produces an embedded tree would improve the exposition.
[Table 1] The last three Suﬀridge polynomials in Table 1 are numerical approximations; it would be helpful to state the precision or to indicate the sense in which they are normalized.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bi-angled tree classification is derived from independent existence and uniqueness arguments; cited prior work supplies supporting lemmas rather than restating the conclusion.

full rationale

The derivation chain is self-contained in the relevant sense. The map from Suﬀridge polynomials to bi-angled trees is defined geometrically (Deﬁnition 2.19), and the converse existence (Theorem 4.1) is proved by an explicit quasiconformal pinching construction starting from f0(z)=z−1/d z^d; no parameter is fitted from the target classification. Uniqueness (Theorem 5.1) is attempted through a pullback argument, and its auxiliary facts—constant conformal curvature, absence of wandering components, cusp/double-point asymptotics—are cited from prior external work [LM14, LLMM18a, LLMM18b], not from the conclusion of Theorem A. The correspondence with CS anti-polynomials (Theorem 6.6) rests on Poirier's external realization and uniqueness theorem for angled Hubbard trees; the present-paper input is the observation that the identity angled tree map with local degree 2 makes a bi-angled tree a CS Hubbard tree. The skeptical issue in Theorem 5.1—the unsupported simultaneous quasiconformal extension Ψ0 matching both Ψ and the Böttcher conjugacy—is a potential correctness gap, not a circular reduction: Lemma 5.4 is not claimed to define Ψ0 by construction, and the missing interpolation is not an equation equating the theorem's output with its input. Self-citations such as [LMM20, §4] for landing of fixed external rays are peripheral supporting facts, not the classification content, and do not make the central claim equivalent to its assumptions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numerical parameters are fitted to data. The paper is pure mathematics and relies on standard tools in quasiconformal analysis and on specific prior classification results, most importantly Poirier's Hubbard tree theory and the authors' earlier lemmas on conformal curvature and Schwarz reflection dynamics.

assumptions (7)

standard math Quasiconformal deformation theory and the measurable Riemann mapping theorem (used to define φt in Propositions 4.3, 4.9, 4.12).
General tool from [BF14] and [Ahl06], accepted background for the pinching constructions.
standard math Sullivan's non-wandering theorem adaptation (used in Proposition 3.4 to rule out wandering Fatou components).
The paper adapts the classical proof and uses finite dimensionality of Σ*_d; the adaptation is summarized, not fully formalized.
domain assumption Poirier's realization and uniqueness theorem for angled Hubbard trees of postcritically finite anti-polynomials (used in Proposition 6.5 and Theorem 6.6).
External classification result from [Poi13, Theorem 5.1] that transfers bi-angled trees to CS anti-polynomials; the paper checks the hypotheses.
domain assumption Constant conformal curvature of f(T) at non-cusp points for f in Σ*_d (Principle 1, from [LM14, Corollary 2.8]).
Used in several places in Section 4 to rule out unwanted self-intersections during pinching; assumes the published lemma from the authors' prior work.
domain assumption Khavinson-Swiatek upper bound of 3d−2 fixed points for harmonic polynomials (Theorem 2.21, from [KS03]).
Used in Propositions 6.2 and 6.3 to count fixed points of CS anti-polynomials.
standard math Warschawski's asymptotic formulas for conformal maps of curvilinear strips (used in Lemma 5.3).
External analytic result applied to strips near cusps and double points.
standard math Ahlfors-Beurling extension theorem and removability of analytic arcs and measure-zero sets (used in Lemmas 5.4 and the proof of Theorem 5.1).
Standard quasiconformal extension and removability results used to promote the droplet homeomorphism to a global map.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Univalent Polynomials and Hubbard Trees." pith.science (2026). https://pith.science/paper/WKIWJTEY

@misc{pith2026190805813,
  author       = {Pith},
  title        = {Pith review of: Univalent Polynomials and Hubbard Trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WKIWJTEY}},
  note         = {Machine review of arXiv:1908.05813}
}

abstract

We study rational functions $f$ of degree $d+1$ such that $f$ is univalent in the exterior unit disc, and the image of the unit circle under $f$ has the maximal number of cusps ($d+1$) and double points $(d-2)$. We introduce a bi-angled tree associated to any such $f$. It is proven that any bi-angled tree is realizable by such an $f$, and moreover, $f$ is essentially uniquely determined by its associated bi-angled tree. This combinatorial classification is used to show that such $f$ are in natural 1:1 correspondence with anti-holomorphic polynomials of degree $d$ with $d-1$ distinct, fixed critical points (classified by their Hubbard trees).

Figures

Figures reproduced from arXiv: 1908.05813 by the authors.

**Figure 1.** The rational map f semi-conjugates the reflection map η of D to the Schwarz reflection map σ of Ω. 2.2. Cusps and Double Points. Definition 2.5. Let Ω ⊂ C be an open set. A boundary point p ∈ ∂Ω is called regular if there is a disc B = B(p, ε) := {z ∈ C : |z − p| < ε} such that Ω ∩ B is a Jordan domain and ∂Ω ∩ B is a simple non-singular real analytic arc; otherwise p is a singular point. Note that if the rational m… view at source ↗

**Figure 2.** Illustrated is Definition 2.19 in which a bi-angled tree T (Ω) is associated to a given unbounded quadrature domain Ω by associating vertices to components of int(C \ Ω) and connecting two vertices by an edge if and only if the corresponding components share a boundary point. We note that this figure is merely meant to illustrate Definition 2.19: Ω was neither explicitly nor numerically computed. Definition 2.18. Tw… view at source ↗

**Figure 3.** Pictured is the dynamical plane of the Schwarz reflection map arising from a Suffridge polynomial in Σ ∗ 4 . The numbers denote the ranks of various tiles. As in Section 2, we define T = Cb \ Ω, T 0 = T \ {The singular points on ∂T}, and T ∞(σ) := [ n≥0 σ −n (T 0 ). We will call T ∞(σ) the tiling set of σ. For any n ≥ 0, the connected components of σ −n(T 0 ) are called tiles of rank n. Note that two distinct tiles … view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Illustrated is the Euclidean rectangle R and the quadrilateral Q = R(z1, z2, w1, w2) of Principle 2. The a-sides of the quadrilateral Q have been labeled. 4.1. The Astroid Case. In this subsection, we prove Theorem 4.1 in the case of the astroid (corresponding to the c…

**Figure 5.** Figure 5: This Figure summarizes Section 4.1. The image of Cb \ D under the map f0(z) = z −1/(3z 3 ) is a quadrature domain Ω0. Q = Cb \Ω0 is mapped conformally to [−1, 1]×[−1, 1] by Ψ. The map Lt(x, y) = (x, y t ) for 1 < t < ∞ determines a non-zero Beltrami coefficient on [−1,…

**Figure 6.** Figure 6: This Figure illustrates the image of the unit circle under the mapping f0. The map f0 has d+1 critical points ξ1, · · · , ξd+1 on the unit circle with corresponding critical values ζ1, · · · , ζd+1. Thus if we suppose, by way of contradiction, the failure of (13), ∂Q∞ …

**Figure 7.** Figure 7: Top: Two non-isomorphic bi-angled trees (for d = 5) whose underlying topological trees are isomorphic. Bottom: No isomorphism between the corresponding augmented trees preserves the circular order of edges meeting at each vertex. The bi-angled tree T and the correspond…

**Figure 8.** Figure 8: This Figure summarizes the first step in the proof of Theorem 4.1. One considers a pair of indices {j, k} ∈ ST and pinches the curve f(Ij ) to the curve f(Ik) by quasiconformally perturbing the quadrilateral Q as in Section 4.1. The same proofs as for Propositions 4.3 …

**Figure 9.** Figure 9: Illustrated is the argument that the modulus of the path family Γt stays bounded away from ∞ as t → ∞. Consider the quadrilateral Qn(ζ tnm , ζtn m+1, ζtn l , ζtn l+1). Let sa(n), sb(n) denote the Euclidean path-distance between the a, b-sides of Qn(ζ tnm , ζtn m+1, ζtn…

**Figure 10.** Figure 10: This Figure summarizes the second step in the proof of Theorem 4.1. Proposition 4.12. With notation as above, there exist a family of quasiconformal maps (φt)t∈[1,∞) : Cb → Cb, and a family of rational maps (ft)t∈[1,∞) such that (φt)z/(φt)z = µt a.e., φt(∞) = ∞, f1 = …

**Figure 11.** Figure 11: Pictured on the left is part of a bi-angled tree T . Also shown is Tb, where only the degree 1 blue vertices of Tb have been illustrated for clarity. In the middle are shown the arcs Jk on T and the pre-images (under φ) of the degree 1 blue vertices of Tb. Elements of…

**Figure 12.** Figure 12: The dynamical plane of the Schwarz reflection map σ associated with the extremal unbounded quadrature domain realizing the unique bi-angled tree with 2 vertices. The limit set of σ is the boundary of the tiling set. Lemma 5.4. Ψ can be extended to a quasiconformal map…

**Figure 13.** Figure 13: Left: A valid configuration of invariant (bounded) immediate basins for a CS anti-polynomial of degree 4 (the non-critical fixed points are marked in red). Right: No CS anti-polynomial of degree 4 can admit such a configuration of invariant (bounded) immediate basins;…

**Figure 14.** Figure 14: The dynamical plane of a critically fixed degree 3 anti-polynomial whose angled Hubbard tree (shown in yellow) is isomorphic to the unique bi-angled tree with 2 vertices. The five repelling fixed points of the anti-polynomial are also marked (in red) We will now deduc…

**Figure 15.** Figure 15: Pictured is the image of T under certain polynomials of degrees 5 and 6 studied in [Suf69]. The proof of Theorem 7.7 uses an approach in [LM14] to “separate” all but the left-most double point, whence the techniques in the proof of Theorem 4.1 apply. For a positive in…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 30 canonical work pages

[1]

L. V. Ahlfors. Lectures on Quasiconformal Mappings (Second Edition) , volume 38 of University Lecture Series . American Mathematical Society, Providence, R.I., 2006

work page 2006
[2]

Aharonov and H

D. Aharonov and H. S. Shapiro. Domains on which analytic functions satisfy quadrature identities. J. Analyse Math. , 30:39--73, 1976

work page 1976
[3]

Branner and N

B. Branner and N. Fagella. Quasiconformal surgery in holomorphic dynamics , volume 141 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, 2014

work page 2014
[4]

Bshouty and A

D. Bshouty and A. Lyzzaik. On C rofoot- S arason's conjecture for harmonic polynomials. Comput. Methods Funct. Theory , 4(1):35--41, 2004

work page 2004
[5]

D. A. Brannan. Coefficient regions for univalent polynomials of small degree. Mathematika , 14:165--169, 1967

work page 1967
[6]

X. Buff. Virtually repelling fixed points. Publicacions Matem \`a tiques , 47:195--209, 2003

work page 2003
[7]

Carleson and T

L. Carleson and T. W. Gamelin. Complex Dynamics . Springer, Berlin, 1993

work page 1993
[8]

V. F. Cowling and W. C. Royster. Domains of variability for univalent polynomials. Proc. Amer. Math. Soc. , 19:767--772, 1968

work page 1968

Show all 31 references

[9]

Douady and J

A. Douady and J. H. Hubbard. \'E tude dynamique des polyn\^omes complexes . Soci \'e t \'e M ath \'e matique de F rance, 2007

2007
[10]

Peter L. Duren. Univalent functions , volume 259 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, New York, 1983

1983
[11]

L. Geyer. Sharp bounds for the valence of certain harmonic polynomials. Proc. Amer. Math. Soc. , 136:549--555, 2008

2008
[12]

K \"o ssler

M. K \"o ssler. Simple polynomials. Czechoslovak Math. J. , 1(76):5--15, 1951

1951
[13]

Khavinson and G

D. Khavinson and G. Swiatek. On the number of zeros of certain harmonic polynomials. Proc. Amer. Math. Soc. , 131:409--414, 2003

2003
[14]

S. V. F. Levy. Critically Finite Rational Maps (Thurston) . ProQuest LLC, Ann Arbor, MI, 1985. Thesis (Ph.D.)--Princeton University

1985
[15]

S.-Y. Lee, A. Lerario, and E. Lundberg. Remarks on W ilmshurst's theorem. Indiana University Mathematics Journal , 64(4):1153--1167, 2015

2015
[16]

S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee. Dynamics of S chwarz reflections: The mating phenomena. https://arxiv.org/abs/1811.04979v2, 2018

2018 arXiv
[17]

S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee. S chwarz reflections and the T ricorn. https://arxiv.org/abs/1812.01573v1, 2018

2018 arXiv
[18]

Lee and N

S.-Y. Lee and N. Makarov . Sharpness of connectivity bounds for quadrature domains . https://arxiv.org/abs/1411.3415, 2014

2014 arXiv
[19]

Lee and N

S.-Y. Lee and N. G. Makarov. Topology of quadrature domains. J. Amer. Math. Soc. , 29(2):333--369, 2016

2016
[20]

Lazebnik, N

K. Lazebnik, N. G. Makarov, and S. Mukherjee. Bers slices in families of univalent maps. https://arxiv.org/abs/2007.02429v2, 2020

2007 arXiv
[21]

Lehto and K

O. Lehto and K. I. Virtanen. Quasiconformal mappings in the plane . Springer-Verlag, New York-Heidelberg, second edition, 1973. Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126

1973
[22]

M. Yu. Lyubich. On typical behavior of the trajectories of a rational mapping of the sphere. Dokl. Akad. Nauk SSSR , 268:22--25, 1983

1983
[23]

C. Michel. Eine B emerkung zu schlichten P olynomen. Bull. Acad. Polon. Sci. S\' e r. Sci. Math. Astronom. Phys. , 18:513--519, 1970

1970
[24]

J. Milnor. Dynamics in one complex variable , volume 160 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, third edition, 2006

2006
[25]

A. Poirier. Hubbard forests. Ergodic Theory and Dynamical systems , 33:303--317, 2013

2013
[26]

D. Sarason. written communication. Feb. 1999, Oct. 2000

1999
[27]

T. J. Suffridge. On univalent polynomials. J. London Math. Soc. , 44:496--504, 1969

1969
[28]

T. J. Suffridge. Extreme points in a class of polynomials having univalent sequential limits. Trans. Amer. Math. Soc. , 163:225--237, 1972

1972
[29]

Sullivan

D. Sullivan. Quasiconformal homeomorphisms and dynamics I . solution of the F atou- J ulia problem on wandering domains. Annals of Mathematics , 122(2):401--418, 1985

1985
[30]

S. E. Warschawski. On conformal mapping of infinite strips. Transactions of the American Mathematical Society , 51(2):280--335, 1942

1942
[31]

A. S. Wilmshurst. The valence of harmonic polynomials. Proc. Amer. Math. Soc. , 126(7):2077--2081, 1998

1998

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

L. V. Ahlfors. Lectures on Quasiconformal Mappings (Second Edition) , volume 38 of University Lecture Series . American Mathematical Society, Providence, R.I., 2006

work page 2006

[2] [2]

Aharonov and H

D. Aharonov and H. S. Shapiro. Domains on which analytic functions satisfy quadrature identities. J. Analyse Math. , 30:39--73, 1976

work page 1976

[3] [3]

Branner and N

B. Branner and N. Fagella. Quasiconformal surgery in holomorphic dynamics , volume 141 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, 2014

work page 2014

[4] [4]

Bshouty and A

D. Bshouty and A. Lyzzaik. On C rofoot- S arason's conjecture for harmonic polynomials. Comput. Methods Funct. Theory , 4(1):35--41, 2004

work page 2004

[5] [5]

D. A. Brannan. Coefficient regions for univalent polynomials of small degree. Mathematika , 14:165--169, 1967

work page 1967

[6] [6]

X. Buff. Virtually repelling fixed points. Publicacions Matem \`a tiques , 47:195--209, 2003

work page 2003

[7] [7]

Carleson and T

L. Carleson and T. W. Gamelin. Complex Dynamics . Springer, Berlin, 1993

work page 1993

[8] [8]

V. F. Cowling and W. C. Royster. Domains of variability for univalent polynomials. Proc. Amer. Math. Soc. , 19:767--772, 1968

work page 1968