Generalized Chebyshev's polinomials over algebraically closed fields
by
Julia Darevskaya
PhD student, Moscow State University

It is known that a plane tree can be reconstructed by the polinomial P over C with no more than two critical values as a preimage of the segment connected its critical values. The polinomial with no more than two critical values is called generalized Chebyshev's polynomial or Shabat's polynomial ( P in ShP(C)). There exists the correspondens between an every isotopic class of trees embedded into C and Shabat's polinomials over C up to the replacement of so called C-equivalency: P(z) --> AP(az+b)+B, A, a, B, b in C , Aa =/= 0, see [1, 2].

Since the definition of Shabat's polinomial is purely algebraical we can consider this concept over more general setting, for example, over allgebraically closed fields or over Z. In the last case under reduction over some primes the polynomials behave differs from its behave in characteristics zero. Such primes we call primes of bad reduction. The finding bad primes for P in ShP(Z) comes to the finding bad primes for any so called uncompressible representative of Q-equivalency class for P. Such a representative exists and can be find in an every class of Q-equivalent Shabat's polynomials.

References

1. Adrianov N.M.; Arothmetic theory of graphs on surfaces. Moskow State University, Ph.D Thesis (1997)[in russian]

2.Shabat G., Zvonkine A.: Plane trees and algebraic numbers. Contemprorary Mathematics, AMS 178 (1994) 233-277.

Date received: January 31, 2001