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Plane trees and Shabat polynomials

✍ Scribed by Jean Bétréma; Alexander Zvonkin

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
538 KB
Volume
153
Category
Article
ISSN
0012-365X

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✦ Synopsis

In his unpublished paper [7] Alexandre Grothendieck has indicated that there exist profound relations between the theory of number fields and that of maps on two-dimensional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.

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