Plane trees and Shabat polynomials Original Research Article

In his unpublished paper [7] Alexandre Grothendieck has indicated that there exist profound relations between the theory of number fields and that of maps on two-dimensional surfaces. This theme was later explored by George Shabat (Moscow) and his students (see [1, 2, 11, 12, 14, 16]). For the simplest class of maps, that of plane trees, this theory leads to a very interesting class of polynomials which generalize Chebyshev polynomials and which we call Shabat polynomials. A catalog of Shabat polynomials for all plane trees up to 8 edges is compiled in [4]. In the present paper we describe the connection between plane trees and Shabat polynomials, give some examples (and counterexamples) and discuss some conjectures.