Harmonic maps, Roskilde, 09-2010

CODY conference

Holomorphic dynamics, Around Thurston’s theorem

On the number of roots of planar harmonic polynomials

report on work of Khavinson-Swiatek and Geyer

Tan Lei, Universit´ e d’Angers

Roskilde, 30.09. 2010

We say thatp(z) =adz^d+· · ·+a1z+a0is aGeyer polynomialif pis real (= all coefficients real),

all critical points aresimple,

≤1realcritical point,

psends each critical pointcto its complex conjugate ¯c.

Theorem (Geyer)

For everyd≥2, there is a Geyer polynomialpof degreed.

This result solved a sharpness problem in the study of harmonic polynomials.

Upper bound for #roots(p(z) − q(z))

p, q two polynomials. Assumed:= deg(p)>deg(q).

Wilmshurst(1998) #roots≤d² (Bezout’s theorem).

In case p(z)−z,

Khavinson-Swiatek(2003) #roots≤3d−2 and the bound is sharp ford= 2,3 (using Fatou’s lemma in complex dynamics).

Remark of Crofoot-Sarason A polynomialpwith only simple critical pointscand withp(c) =cwould realize the sharp bound.

Bshouty-Lyzzaik(2004) Such polynomial exists ford= 4,5,6,8 (using algebraic method not generalizable)

Geyer(2008)sharpness for∀d.

Upper bound for #roots(p(z) − q(z))

p, q two polynomials. Assumed:= deg(p)>deg(q).

Wilmshurst(1998) #roots≤d² (Bezout’s theorem).

In case p(z)−z,

Khavinson-Swiatek(2003) #roots≤3d−2 and the bound is sharp ford= 2,3 (using Fatou’s lemma in complex dynamics).

Remark of Crofoot-Sarason A polynomialpwith only simple critical pointscand withp(c) =cwould realize the sharp bound.

Bshouty-Lyzzaik(2004) Such polynomial exists ford= 4,5,6,8 (using algebraic method not generalizable)

Geyer(2008)sharpness for∀d.

Upper bound for #roots(p(z) − q(z))

p, q two polynomials. Assumed:= deg(p)>deg(q).

Wilmshurst(1998) #roots≤d² (Bezout’s theorem).

In case p(z)−z,

Khavinson-Swiatek(2003) #roots≤3d−2 and the bound is sharp ford= 2,3 (using Fatou’s lemma in complex dynamics).

Remark of Crofoot-Sarason A polynomialpwith only simple critical pointscand withp(c) =cwould realize the sharp bound.

Bshouty-Lyzzaik(2004) Such polynomial exists ford= 4,5,6,8 (using algebraic method not generalizable)

Geyer(2008)sharpness for∀d.

Upper bound for #roots(p(z) − q(z))

p, q two polynomials. Assumed:= deg(p)>deg(q).

Wilmshurst(1998) #roots≤d² (Bezout’s theorem).

In case p(z)−z,

Khavinson-Swiatek(2003) #roots≤3d−2 and the bound is sharp ford= 2,3 (using Fatou’s lemma in complex dynamics).

Remark of Crofoot-Sarason A polynomialpwith only simple critical pointscand withp(c) =cwould realize the sharp bound.

Bshouty-Lyzzaik(2004) Such polynomial exists ford= 4,5,6,8 (using algebraic method not generalizable)

Geyer(2008)sharpness for∀d.

Upper bound for #roots(p(z) − q(z))

p, q two polynomials. Assumed:= deg(p)>deg(q).

Wilmshurst(1998) #roots≤d² (Bezout’s theorem).

In case p(z)−z,

Khavinson-Swiatek(2003) #roots≤3d−2 and the bound is sharp ford= 2,3 (using Fatou’s lemma in complex dynamics).

Remark of Crofoot-Sarason A polynomialpwith only simple critical pointscand withp(c) =cwould realize the sharp bound.

Bshouty-Lyzzaik(2004) Such polynomial exists ford= 4,5,6,8 (using algebraic method not generalizable)

Geyer(2008)sharpness for∀d.

Counting roots for f = p − q, d = deg(f )

Lower bound and surjectivity.

degp6= degq=⇒ ∀w∈C, #f⁻¹(w)≥d .

Geyer(again) ∀d≥2,

∃degree-d p(z)∈R[z]

≤1 critical points real







s.t. ∃w∈C, #{p(z)−z=w}= 3d−2.

Theorem (El Amrani-Loeb-T. 2010) ∀d≥2,

∀ degree-d p(z)∈R[z]

all critical points real







∀w∈R, #{p(z)±z=w} ≤







2d deven

2d+ 1 dodd.

And the bound is sharp for everyd.

Proof of Geyer’s theorem

Findp∈R[z] such thatp⁰(c) = 0 =⇒











p⁰⁰(c)6= 0

c6= ¯c (except possibly one) p(c) =c

Important Notice : Every critical point is 2-periodic. Apply Thurston’s theorem :

1. Create a topological (C^∞, quasi-regular) polynomialG. 2. Prove that6 ∃ obstructions (Levy).

3. Apply Thurston’s theorem to obtain a polynomialP. 4. Study the symmetry ofP.

Proof of Geyer’s theorem

Findp∈R[z] such thatp⁰(c) = 0 =⇒











p⁰⁰(c)6= 0

c6= ¯c (except possibly one) p(c) =c

Important Notice : Every critical point is 2-periodic.

Apply Thurston’s theorem :

1. Create a topological (C^∞, quasi-regular) polynomialG. 2. Prove that6 ∃ obstructions (Levy).

3. Apply Thurston’s theorem to obtain a polynomialP. 4. Study the symmetry ofP.

Proof of Geyer’s theorem

Findp∈R[z] such thatp⁰(c) = 0 =⇒











p⁰⁰(c)6= 0

c6= ¯c (except possibly one) p(c) =c

Important Notice : Every critical point is 2-periodic.

Apply Thurston’s theorem :

1. Create a topological (C^∞, quasi-regular) polynomialG.

2. Prove that6 ∃ obstructions (Levy).

3. Apply Thurston’s theorem to obtain a polynomialP. 4. Study the symmetry ofP.

Proof of Geyer’s theorem

Findp∈R[z] such thatp⁰(c) = 0 =⇒











p⁰⁰(c)6= 0

c6= ¯c (except possibly one) p(c) =c

Important Notice : Every critical point is 2-periodic.

Apply Thurston’s theorem :

1. Create a topological (C^∞, quasi-regular) polynomialG.

2. Prove that6 ∃ obstructions (Levy).

3. Apply Thurston’s theorem to obtain a polynomialP. 4. Study the symmetry ofP.

Proof of Geyer’s theorem

Findp∈R[z] such thatp⁰(c) = 0 =⇒











p⁰⁰(c)6= 0

c6= ¯c (except possibly one) p(c) =c

Important Notice : Every critical point is 2-periodic.

Apply Thurston’s theorem :

1. Create a topological (C^∞, quasi-regular) polynomialG.

2. Prove that6 ∃ obstructions (Levy).

3. Apply Thurston’s theorem to obtain a polynomialP.

4. Study the symmetry ofP.

Proof of Geyer’s theorem

Findp∈R[z] such thatp⁰(c) = 0 =⇒











p⁰⁰(c)6= 0

c6= ¯c (except possibly one) p(c) =c

Important Notice : Every critical point is 2-periodic.

Apply Thurston’s theorem :

1. Create a topological (C^∞, quasi-regular) polynomialG.

2. Prove that6 ∃ obstructions (Levy).

3. Apply Thurston’s theorem to obtain a polynomialP.

4. Study the symmetry ofP.

We want to prove that P is real.

Letπ:z7→z.¯

Setφ⁰₁=π◦φ1◦π⁻¹,φ⁰₀=π◦φ0◦π⁻¹, andF =π◦P◦π⁻¹. Then F(z) =P(¯z) is again a polynomial and we have the following chains of commutative diagrams:

Cb

G

φ⁰₁

##

Cb

oo π ^φ1 //

G

Cb

P₁

π //

Cb

F

Cb

φ⁰₀

;;

Cb

oo π

φ0

// bC ^π //

Cb.

Cb

G

φ⁰₁

##

Cb

oo π ^φ1 //

G

Cb

P1

π //

Cb

F

Cb

φ⁰₀

;;

Cb

oo π

φ0

// bC π //

Cb.

Due the unicity ofψmakingφ0◦G◦ψ⁻¹ holomorphic, φ0 is real =⇒





 φ₁ P1

are real =⇒





 φ_n Pn

are real,

where (φn, Pn) are the pairs that appear in the Thurston algorithm.

ButP_n→P, so isP.

Geometry and counting of Geyer polynomials

Geyer:

#{degreedGeyer polynomials}/(real conj.)

=m−th Catalan number = 1 m+ 1



 2m

m



 if







d= 2(m+ 1) d= 2m+ 1 Thurston’s theorem−→Classification results.

Why studying p(z) − q(¯ z)? p(z) − z? ¯

• real algebraic geometry

• global singularity theory of planar maps (local study by Wittney)

• holomorphic dynamics applies

• gravitational lensing (astrophysics)

The map z 7→ z

− z, cr. set and cr. value set

-2.4 -1.6 -0.8 0 0.8 1.6 2.4

-0.8 0 0.8

Case z

− z, co-critical set

-2.4 -1.6 -0.8 0 0.8 1.6 2.4

-0.8 0 0.8

Cubic polynomials

El Amrani-Loeb-T.(work in progress)

• f(z) =q(z)−z, degq= 3 ∼ z³

3 +pz−z .

• Cf =











a Jordan curve if|p|<1

figure 8 if|p|= 1

∂(Ω¹tΩ²Jordan domains) if|p|>1

• |p|>1 =⇒universality ofz²−z), and

• #f⁻¹(c) =















7 |c∈f(Ω¹)∩f(Ω²) 5 |c∈f(Ω¹)Mf(Ω²) 3 |c∈Crb f(Ω¹)∪f(Ω²)

•#{Whitney’s cusp singularities}=8,4,6

Upperbound #{p(z) − z} ≤ 3d − 2}

SetK(z) =p(z),f(z) =K(z)−z.

{K(z) =z,|K⁰|>1} {K(z) =z,|K⁰|<1}

k k

#{f(z) = 0}= #{f(z) = 0,deg_z=−1} + #{f(z) = 0,deg_z= 1}

degree

= d+ #{deg_z= 1}

= d+ 2·#{attracting fixed pts ofK}

Fatou

≤ d+ 2(d−1) = 3d−2