Combinatorics of polynomials, Shanghai, 04-2013

Seminar at Fudan university 22 April 2013

Combinatorics of complex polynomials

Tan Lei Université d’Angers

About polynomials

Atci,P⁰(ci) =0. They are the critical pointsofP. The valuesv_i =P(c_i) are the critical values ofP.

Main question : Can one use thec_i’s or the v_i’s to determine the polynomial?

Prescribing critical points

Given an (unordered) listX of d −1 points in C(might not be distinct), one can find a polynomial of degreed realizingX as the set of critical points :

P⁰(z) =Q

c∈X(z−c), P(z) = Z z

0

P⁰(z)dz . One can actually find all such polynomials :

Q(z) =A Z z

0

P⁰(z)dz +B =φ◦P(z), φ(z) =Az+B A∈C^∗,B∈C.

C

P . &Q

C −→

φ C

Prescribing critical points

Given an (unordered) listX of d −1 points in C(might not be distinct), one can find a polynomial of degreed realizingX as the set of critical points :

P⁰(z) =Q

c∈X(z−c), P(z) = Z z

0

P⁰(z)dz . One can actually find all such polynomials :

Q(z) =A Z z

0

P⁰(z)dz +B =φ◦P(z), φ(z) =Az+B A∈C^∗,B ∈C.

C

P . &Q

C −→

φ C

Prescribing critical values

Existence? How many? Explicit formulae?

c_1 c_3

v_1 v_3 c_2

v_2

-2 -1 1

-0.5 0.5 1.0 1.5

Prescribing critical values

Existence? How many? Explicit formulae?

Two polynomialsf,g :C→Care said to beisomorphic (or covering equivalent) if there is an affine mapM(z) =az+b, a6=0, such that C →^M C

g & .f C

. Clearly f andg share the same degree and the same critical value setV.

So how manyisomorphism classesare there ?

How many ? In the simple critical values case :

1. (Lyashko-Looijenga) For V a set ofd −1 distinct points in C, the number of isomorphism classes withV as the critical value set is equal to d^d−3:

1,1,1,4,25,216,2401,· · · .

2. Theorem LL For V a set of d−1distinct points in C, the number of monic centered polynomials with V as the critical value set is equal to d^d−2:

1,2,3,16,125,756,16807,· · · .

3. Real polynomials with real critical points and simple critical values : with generating function secx+tanx :

1,1,1,2,5,16,61,272,· · ·.

In particular any set of real points can be realized as critical values of a real polynomial.

How to distinguish?

1. No explicite formulae!!

2. Ford −1 distinct real pointsV ={A₁ <A₂<· · ·<A_d−1}, and anyzig-zag permutation

σ =

1 2 3 · · · d−1

σ(1) σ(2) σ(3) · · · σ(d −1)

such thatσ(d−1)< σ(d −2)> σ(d −3)< σ(d−4)· · ·,

∃!degreed monic-centered real polynomialf_V,σ with critical points c₁<c₂ <· · ·<c_d−1 so thatv_i :=f(c_i) =A_σ(i).

So the set ofσ’s provides a universal encodingof these real polynomials.

3. In degree 4, givenA<B <C,∃ two possibilities :

v₁ =A<B =v₃ andv₂=C or v₃=A<B =v₁ andv₂ =C. In each case there is a unique real monic centered polynomial realizing the configuration.

Use parking functions (inspired by Tomasini)

Setn=d−1. It is known :

Theorem(n+1)ⁿ⁻¹=d^d−2 is the number of parking functions of n cars on n parking spots

it is also the number of (non-embedded) rooted trees whose edges are labeled by c₁,· · ·,c_n.

Aparking functionis a functionp :{1,2,· · ·,n} → {1,2,· · · ,n}

such that#p⁻¹({1,2,· · · ,i})≥i, ∀i =1,· · ·,n.Here is an example :

cars 1 2 3 4 5 6 7 8 9

favorite parking spotsp(i) 5 5 2 1 5 1 7 3 9 . This is a necessary and sufficient condition for thecivilized parking algorithmto work. There is also a wild parking algorithm.

The round about argument of Pollack gives an easy proof about the number of these functions.

∃a bijection p ↔(σ_civilized, σ_wild) (Poulalhon).

A bijection between parking functions and trees

A parking function can be represented fiber wisely as follows (each fiber should be displaced in increasing order from left to right ) :

cars p⁻¹(1) p⁻¹(2) · · · p⁻¹(n)

favorite parking spots 1 2 · · · n

Here is our example again:

cars 1 2 3 4 5 6 7 8 9

favorite parking spotsp(i) 5 5 2 1 5 1 7 3 9 .

cars 4 6 3 8 1 2 5 7 9

favorite parking spots 1 2 3 4 5 6 7 8 9 .

For the cars to be all parked at leasti cars should prefer to park the firsti spots. That is :

#p⁻¹({1,2,· · ·,i})≥i, ∀i =1,· · · ,n.

From a parking function to a tree

Arooted labeled tree is a non-embedded tree with marked vertex as root, and with edges labeled asc₁,· · · ,c_n.

Write down one by one the consecutive car numbers in the list p⁻¹(1),p⁻¹(2),· · ·, p⁻¹(n), to obtain a list q(1),q(2),· · · ,q(n).

And then make a rooted tree so that the descendants edges of the root arec_p−1(1), and those of c_q(i−1) arec_p−1(i) (no descendent if the set is empty).

the root and the edges ∗ q(1) · · · q(n−1) q(n) cars, descendant edges p⁻¹(1) p⁻¹(2) · · · p⁻¹(n)

favorite parking spots 1 2 · · · n

. In the above example,

root and edges ∗ 4 6 3 8 1 2 5 7 9

cars, descendants edges 4 6 3 8 1 2 5 7 9

favorite parking spots 1 2 3 4 5 6 7 8 9

From trees to parking functions

From a tree with root∗and with edges c1· · ·,cn, we can make a breadth first search starting from∗: we build a queue, each time putting the descendant edges of the new queue-head to the end of the queue, in increasing order if there are several descendant edges.

The favorite parking spot corresponds to the entering order of the group of descendant edges to the queue.

heads of search ∗ 4 6 3 8 1 2 5 7 9

cars, descendant edges 4 6 3 8 1 2 5 7 9

entering order of desc. 1 2 3 4 5 6 7 8 9

favorite parking spots

Proof of LL, bijection between trees to monic centered polynomials

Embed the tree in the plane by increasing order around each vertex.

Then add extra edges at each vertex so that each vertex hasn edges and they are increasingly ordered from 1 to n.

This tree is now the union ofd stars joint at n=d −1 points c_v,v ∈ {1,· · ·,n}. In the star centered at the root there is a unique edge from the root to the edge 1. Orient this edge from the root.

A setV ofn distinct points on the unit circle, defines a star S in D centered at 0, withn radial lines as branches, naturally ordered from 1 ton, following their angles in[0,2π[.

Now construct a topological polynomial sending each star toS preserving the cyclic order. Uniformize it so that a ray landing on the right of the marked edge is sent to[1,+∞[. This determines a unique monic centered polynomial with critical value setV.

Conversely

Any monic centered polynomialP with critical value setV ⊂S¹ pulls backS to a planar treeT. We may callc_v the critical point that is mapped to thev-th critical value. ThenT isd copies ofS joint atc₁,· · · ,c_n. The ray[0,+∞[ has a unique preimage that is asymptotic to the positive real axis at∞. Assign the star-center on the left of its landing point to be the root ofT.

This bijection gives a combinatorial proof of Theorem LL.

The proof isNOT constructive, no formulae for the coefficients ! The set of parking functions gives anon universal encoding of the polynomials.

Conversely

Any monic centered polynomialP with critical value setV ⊂S¹ pulls backS to a planar treeT. We may callc_v the critical point that is mapped to thev-th critical value. ThenT isd copies ofS joint atc₁,· · · ,c_n. The ray[0,+∞[ has a unique preimage that is asymptotic to the positive real axis at∞. Assign the star-center on the left of its landing point to be the root ofT.

This bijection gives a combinatorial proof of Theorem LL.

The proof isNOT constructive, no formulae for the coefficients ! The set of parking functions gives anon universal encoding of the polynomials.

Algebraic proof of Theorem LL, computing the coefficients

Set

LL:f 7→ LL_f, LL_f(t) = Y

vcritical values off

(t−v).

Note thatLLsends a degree d monic centered polynomial inz to a degree d-1 monic polynomial int.

Notice that if we replacef byf −w, we will move all critical values by a translation by−w. So

LL_f−w(t) = Y

ucritical value off−w

(t−u) =

Y

vcritical value off

(t−(v −w)) =LL_f(t+w).

So for a given vectorc∈C^d−1, centered, we may assign a unique polynomialP_c with critical point setcand with critical value set centered. We setLL(c) =c v to be the corresponding critical value vector of this uniqueP_c.

TheoremLL(λc) =c λ^dLL(c)c (it is a homogenous polynomial of degree d per each coordinate), it is proper, in particular open and surjective. And it is of multiplicity d^d−2.

An example : in degree 4 :

LLc :



 c₁ c₂ c₃



7→ 1 3









−c₁⁴+2c₁³(c₂+c₃)−6c₁(c₁c₂c₃)

−c₂⁴+2c₂³(c₁+c₃)−6c₂(c₁c₂c₃)

−c₃⁴+2c₃³(c₁+c₂)−6c₃(c₁c₂c₃)









−center

P=

ci=0

−







c₁⁴+2c₁(c₁c₂c₃) c₂⁴+2c₂(c₁c₂c₃) c₃⁴+2c₃(c₁c₂c₃)





+1 3







c₁⁴+c₂⁴+c₃⁴ c₁⁴+c₂⁴+c₃⁴ c₁⁴+c₂⁴+c₃⁴







The proof (inspired by LL and Douady-Sentenac)

Setc_i^∗=λci. In degree 4, set g^∗(x) =4(x −c₁^∗)(x−c₂^∗)(x −c₃^∗).

Sog^∗(λx) =λ³4(x −c₁)(x −c₂)(x −c₃) =λ³g(x)and g^∗(λx) =λ^d−1g(x) in general.

v_i^∗ =



 Z c_i^∗

0

− 1

d −1

d−1

X

j=1

Z c_j^∗ 0



g^∗(u)du

=



 Z λci

0

− 1

d −1

d−1

X

j=1

Z λcj

0



g^∗(u)du

=



 Z c_i

0

− 1

d−1

d−1

X

j=1

Z c_j 0



g^∗(λx)λdx =

= λ^d



 Z ci

0

− 1

d −1

d−1

X

j=1

Z cj

0



g(x)dx=λ^dv_i .

Properness

Now we need to determineLLc⁻¹(0). Notice that all critical values off are at 0 thenf is necessarily(z−c)^d. Making f centered means thatc =0 andf(z) =z^d. So all critical points off are also at 0. ThereforeLLc⁻¹(0) ={0} andLLc is proper.

Now one can apply Bezout’s theorem to conclude thatLLc is of multiplicityd^d−2 from the d−2-dimensional hypersurface Pc_i =0 to thed −2-dimensional hypersurfaceP

v_i =0.

Given now an ordered listV of d−1 points in C, here is how to get all monic centered polynomials realizingV as critical value set:

V give an order

−→ v^centering−→ v−C ←−^LLc

d^d−2:1

cforget order

−→ P_c−→P_c+C =f_a. This proof ismore constructive, but gives no encoding. We will see that

a c

→v is a covering over the set of v with pairwise distinct coordinates.

Analytic continuation, the scilab code of H.H. Rugh.

For a vectora=





 a_d

... a2





∈C^d−1, consider a monic centered polynomialfa(z) =a_d+ad−1z+· · ·+a₂z^d−2+z^d. Theorem. The set

W =









 a c v



∈C^3(d−1)

c=critical points of f_a v=f_a(c)







is analgebraic set. The projectionπv :W →C^d−1 is a finite degree covering over the subsetV ofv with pairwise distinct coordinates.

(We have used a short hand notation that for a function φ:C→C, the formulaφ(c) denotes the column vector (φ(c1),· · · , φ(cd−1))^t.)

One can then expect to lift paths realizing prescribedv.

The algebraic set and the non-autonomous ODE

Proof. Write in inner product form f_a(z) = 1 z · · · z^d−2

·a+z^d =B₀(z)·a+z^d g_a(z) :=f_a⁰(z) = 0 1 · · · (d −2)z^d−3

·a+dz^d−1= B₁(z)·a+dz^d−1

whereB₀(z),B₁(z) denote the horizontal vectors in the formulae.

For anyc∈C^d−1 considerF :C^2(d−1) →C^2(d−1) : a

c

7→

f_a(c) g_a(c)

=

B₀(c) B₁(c)

·a+ c^d

dc^d−1

.

ThenW =









 a c v



∈C^3(d−1) F

a c

− v

0

= 0

0





 .

The jacobian, and the Implicit Function Theorem

At a point



 a c v



∈W, the jacobian matrix of F

a c

− v

0

=

f_a(c)−v g_a(c)

takes the form

A O −Id

C D O

with O the (d −1)×(d −1)zero matrix, D the diagonal matrix with diagonal entriesf_a⁰⁰(c₁),· · · ,f_a⁰⁰(c_d−1) and

A=













1 c1 · · · c₁^d−2 1 c₂ · · · c₂^d−2 ... ... ... 1 c_d−1 · · · c_d−1^d−2











 , D =











f_a⁰⁰(c₁) 0 · · · 0 0 f_a⁰⁰(c₂) · · · 0

... ... ...

0 0 · · · f_a⁰⁰(c_d−1)









 .

If thec_i’s are pairwise distinct, on one hand the Van der Monde matrixAis invertible, on the other hand all critical points of f_a are simple sof_a⁰⁰(c_i)6=0. ThereforeA,D and

A O

C D

are invertible.

The scilab program CV-05.sci lifts any paths within a coordinatev_i of the covering overV, and projects the path to the c-coordinates.

Given a smooth pathv(t)⊂ V and



 a(0) c(0) v(0)



∈W, the corresponding path

a(t) c(t)

is the solution of the following non-autonomous ODE:

A(a,c) O C(a,c) D(a,c)

˙ a c

= v(t)˙

0

or ˙

a c

=

A(a,c) O C(a,c) D(a,c)

−1 v(t)˙

0

.

The scilab program showsc(t) when the mouse describesv(t).

Pairing critical points and critical values

LemmaThe set of pairs (critical point, its critical value) of a rational map f determines uniquely f . In other words two maps sharing the same set are identical.

For polynomials, the set of pairs (critical point, its critical value) can be represented by one straight segment inClinking each pair of points. This is what CV-05.sci and CV-web-02.sci show.

The program CV-web-02.sci shows the monodromy action, when moving the critical values, of the tree encoding.

Relation with holomorphic dynamics

Implementing Thurston’s algorithm...