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*Generalized Chebyshev polynomials* have been introduced in

**G.B.Shabat and I.A.Voevodskii**- Drawing curves over number fields
*The Grothendieck Festschift*, vol.3, Birkhäuser, 1990, pp. 199-227.

In this brief introduction I reproduce only a few facts from this beatiful theory; these facts should be sufficient for understanding what I have done. More information can be found via WWW from site of J.Betrema and from thesis of Louis Granboulan. I also recommend survey paper

**G.Shabat and A.Zvonkin**- Plane trees and algebraic numbers
*Contemporary Math*., 1994, vol.178, pp.233--275.

*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

*P*'(*z*)=0 =>*P*(*z*)=*A*or*P*(*z*)=*B*

EXAMPLE. The classical Chebyshev polynomials of the first kind

*T*(_{n}*z*)=cos(*n*arccos(*z*))

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.

FACT 1. The inverse image

The first natural question: * what trees can be obtained in this way?*

FACT 2. Every tree is the inverse image

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.

FACT 3. A plane tree determines corresponding Chebyshev polynomial uniquely up to linear transformations.

Thus there is a striking natural one-to-one correspondence between plane trees and classes of linear equivalent generalized Chebyshev polynomials.

FACT 4. Among generalized Chebyshev polynomials representing given plane tree there is always a polynomial all coefficients of which are algebraic numbers.

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 | ||
---|---|---|---|

"True" picture:
.gif .ps.gz .dvi .tex |
"True" picture:
.gif .ps.gz .dvi .tex | ||

The polynomial: .ascii .gif .ps.gz .tex .nb | |||

Comments |

The polynomial was found by a program written in MATHEMATICA.

These polynomial and pictures were put on WWW on March 10, 1998.

Tree M11 | |
---|---|

Polynomial: .gif .ps.gz .tex .nb | |

"True" picture: .gif .ps.gz .dvi .tex | |

Comments |

The polynomial was found by a program written in MAPLE.

These polynomial and picture were put on WWW on January 20, 1998.