Generalized Chebyshev polynomials and plane trees

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Generalized Chebyshev polynomials and plane trees

• Anton Bankevich
• St. Petersburg State University
• Jass07

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Chebyshev
Pafnuty Lvovich Chebyshev( May 26, 1821
December 8, 1894) One of nine children, he was
born in the village of Okatovo, the district of
Borovsk, province of Kaluga into the family of
landowner Lev Pavlovich Chebyshev. In 1832 the
family moved to Moscow. In 1837 Chebyshev started
the studies of mathematics at the philosophical
department of Moscow University and graduated
from the university as the most outstanding
candidate.
In 1847, Chebyshev defended his dissertation
About integration with the help of logarithms
at St Petersburg University. Chebyshev lectured
at the university from 1847 to 1882. In 1882 he
left the university and completely devoted his
life to research. Chebyshev is known for his work
in the field of probability, statistics and
number theory. Chebyshev is considered to be one
of the founding fathers of Russian mathematics.
Among his students were Aleksandr Lyapunov and
Andrey Markov
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Chebyshev polynomials
There exist a lot of definitions of Chebyshev
polynomials. We shall now consider several of
them.
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Minimal norm
For any given n, among the polynomials of degree
n with the leading coefficient 1, Tn/2n-1 is the
one in which the maximal absolute value on the
interval - 1,1 is minimal. This maximal
absolute value is 1/2n-1, and Tn(x) reaches
this maximum exactly n 1 times with x -1, 1
and with the other n - 1 extremal points of f.
Chebyshev polynomials are important in the
approximation theory because the roots of the
Chebyshev polynomials (which are also referred to
as Chebyshev nodes) are used as nodes in
polynomial interpolation.
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Basis of the space of polynomials

• Chebyshev polynomials appear to be an orthonormal
  basis for the space of polynomials in one
  variable with respect to the following scalar
  product.

So, for any polynomial f(x) of degree n the
coefficients in the formula can be found in the
following way
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Formulas for Chebyshev polynomials
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Composition

• Th Chebyshev polynomials form an Abelian group
  with respect to the operation of composition

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Generalizations

• A lot of attempts were made to generalize the
  notion of Chebyshev polynomials.

Polynomials in several variables
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Motivation
Let f(z) be a polynomial of degree n with complex
coefficients Let us consider an inverse image of
a random point w
Usually, this set consists of n distinct points.
What is the minimal size of
the inverse image of the point w?
The answer is 1. The only examples are a (bcz)n.
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Motivation
And what is the minimal common size of the
inverse image of two points w1 and w2?
The answer is n1. But why?
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Motivation minimal size of inverse image
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Motivation
But when does this degenerate case take place?
What are the examples?
An obvious example is a polynomial zn,where w10,
w2 is a nonzero complex number. But there are
some more examples.
The other example is Tn(z) Chebyshev polynomials,
where w1-1, w21. It can be easily seen that the
inverse image of these points is
.
But there are much more examples of this kind.
And they are called generalized Chebyshev
polynomials.
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Motivation minimal size of inverse image
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Definition
A complex point z is called a critical point of
polynomial f if f(z)0.
A point w is called critical value of f if there
exists a critical point z such that f(z)w.
Polynomial f(z) is called generalized Chebyshev
polynomial (GCP) if it has at most two critical
values. These polynomials are sometimes called
Shabat polynomials
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Certain properties of GCP
The following can be easily seen
If f(z) is a GCP, then f(azb) is GCP
If f(z) is a GCP, then af(z)b is GCP
We shall call polynomials f and g equivalent if
there exist constants a, b, A, B, such that
All the polynomials equivalent to certain GCP
appear to be GCP too. So since this moment we
shall observe only the polynomials with critical
values equal to 0 and 1.
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Geometric point of view
Let us consider the inverse image of a segment.
We colored the ends of the segment in black and
white colors. Let C0 C1 be the segment free of
critical values of f. Then inverse image would be
a set of disjoint sets, each of them being
homeomorphic to a segment.
f
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Geometric point of view
Then we suppose that there are no critical points
inside a segment, but they may appear at one or
both ends. Then some curvilinear segments may
glue together with the monochrome vertexes.
f
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Inverse image of a segment
But what picture can we see in the case where f
is GCP?
The answer is that we will see a bicolored plane
tree
It is a star for zn
- and a chain for Chebyshev polynomials
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Inverse image of a segment
Th For each GCP f the inverse image of a segment
is a bicolored plane tree.
Proof In case of existence of a circuit our
polynomial g takes only real values on the
boundary of domain bounded by our circuit.
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Proof of theorem
Then in case of existence of a circuit our
polynomial g takes only real values on the
boundary of domain bounded by our circuit.
Then the harmonic function Im(g) equals zero on
the boundary.
But it means that Im(g)0 on the whole domain,
which is a contradiction.
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The inverse theorem
Th For each bicolored plane tree T there exists a
GCP such that the corresponding inverse image is
T. Moreover, this GCP is unique up to the
equivalence introduced above.
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Plane tree
What is plane tree?

• Plane tree is a picture of tree

• Plane tree is a structure of a tree and for
  every vertex v a cyclic order on the vertices
  adjacent to v.

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Canonical geometric form
For each bicolored plane tree, there exists a
GCP. And we have presented a way to construct a
geometrical form from the GCP. So for each tree
we can construct its canonical geometric form.
Once more
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Examples of geometric forms
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Examples of geometric forms
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Construction of GCP
The next question is Given a tree, how can we
construct a polynomial?
The first way is to write a system of equations
ai and ßi are the degrees of black and white
vertices of the graph.
Now we have n algebraic equations in n2
variables. This is sufficient to construct the
polynomial.
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Tree families
While constructing our GCP, we only used ai and
ßi, which are the degrees of black and white
vertices of the graph.
But for each multiset of degrees there are a lot
of trees with the same multisetset. All such
trees form a family lta, ßgt, where a(a1, a2,
ak), ß (ß1, ß2, ßn1-k).
By solving the equations from the previous slide
we can find all the polynomials which correspond
to the trees of this family.
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Several trees and their polynomials
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Several trees and their polynomials
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Computation of GCP
The computation of coefficients of GCP for a tree
with a big number of vertices becomes a
complicated problem since we have to solve a
system of algebraic equations of a high degree,
so another way had to be found and it has been
found.
The idea of the new method is the
following First we calculate coefficients not
precisely. It can be done by using the class of
polynomials with the following conditions There
exist complex numbers C and w such that g(w)
C, f(w)0, f(w)?0 If f(z) 0, then f(z) 1
or z w
In the second step we calculate precise values of
the coefficients.
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Computation of GCP a small example
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Galois group action on trees
It can be easily seen that the group Gal(Q/Q)
acts on the set of Chebyshev polynomials with
algebraic coefficients.
So we can define an action of Gal(Q/Q) of
bicolored plane trees.
This leads us to the definition of the field of
definition of a tree this is a field which
corresponds to the subgroup of Gal(Q/Q) that
fixes the given tree.
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GCP coefficients
But what is the connection between the field of
definition and generalized Chebyshev polynomials?
The answer is given in the following theorem
Th For any bicolored plane tree there exists a
generalized Chebyshev polynomial whose
coefficients belong to the field of definition of
the tree.
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Orbits vs families
It can be easily seen that the action of Galois
group on the tree does not change the multiset of
valences of the tree. So the orbit of a tree is
a subset of its family. But do they coincide?
The answer is not always.
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Orbits vs families
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Composition of polynomials
It is well known that the composition of
Chebyshev polynomials is Chebyshev polynomial,
too. And is this the case for generalized
Chebyshev polynomials?
Th Let f, g be GCP such that f(0), f(1) lie in
0, 1. Then the composition f(g(x)) is also GCP.
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Composition of trees
Let us imagine that we have two trees T1 and T2.
Their polynomials are f1 and f2. And we need to
construct a tree which corresponds to f1?f2. Do
we have to calculate f1 and f2, find their
composition and then construct a tree or there is
a direct way?
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Composition of trees
head
body
tail
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Composition of trees
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Plane maps Belyi functions
Till this moment we have been talking about
plane trees, but what about plane maps?

• Let f be a rational function on the Riemann
  sphere that satisfies the following conditions
• f has only three critical values 0, 1, infinity
• All points of f-1(1) are critical with the
  degree exactly equal to 2.

Such functions are called Belyi functions.
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Plane maps map construction
But how are Belyi functions connected with maps?
f-1(8) are vertices.
f-1(1, 8) are edges ( f-1(1) are points on edges )
f-1(0) are faces