Generalized Chebyshev polynomials have been introduced in
Generalized Chebyshev polynomial (Shabat polynomials) can be defined as polynomials (in one complex variable with complex coefficients) for which there are two different complex numbers A and B (called the critical values) such that
The classical Chebyshev polynomials arose as a solution to the problem
of finding a function with the least deviation from zero.
Generalized Chebyshev polynomials are of interest by different reasons.
Namely, let us look at a generalized Chebyshev polynomial P of degree n as a map from the complex plane into itself.
Let [A,B] denote the segment of a straight line with A and B as the end points.
The first natural question: what trees can be obtained in this way?
After such a nice answer, second question arises: how many polynomials can generate given tree?
It is easy to see that if P is a generalized Chebyshev Polynomial,
then so is polynomial CP(cz+d)+D, moreover, it
represents the same tree (of course, provided that both C and c
are different from zero).
In some natural sense these two linear transformations exhaust the variety of polynomials representing given tree.
Namely, every drawing of a tree on the plane introduces an additional structure--circular order of edges around given vertex (say, clock-wise).
Dealing with Chebyshev polynomials, it is natural to speak about plane trees understanding by them trees with this additional structure.
Thus there is a striking natural one-to-one correspondence between plane trees and classes of linear equivalent generalized Chebyshev polynomials.
A catalog of generalized Chebyshev polynomials for trees with small number of edges was constructed by J.Betrema.
Unfortunately, computational complexity grows quickly with the growth of the number of edges. To a great extend this is due to the growth of the smallest algebraic field containing the coefficients of the polynomial.
I wrote programs which first calculates the coefficients numerically with great precision. Then by well-known technique exact algebraic numbers can be found provided that the accuracy is large while the degree of numbers is small.
I will put on WWW trees and corresponding polynomials found by these programs.
|Formats of files|
|Tree M23a||Tree M23b|
.gif .ps.gz .dvi
.gif .ps.gz .dvi
|The polynomial: .ascii .gif .ps.gz .tex .nb|
|Polynomial: .gif .ps.gz .tex .nb|
|"True" picture: .gif .ps.gz .dvi .tex|