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On some algebraic properties of generalized Chebyshev polynomials. (English. Russian original) Zbl 1159.12307

Russ. Math. Surv. 58, No. 1, 175-176 (2003); translation from Usp. Mat. Nauk 58, No. 1, 181-182 (2003).
Generalised Chebyshev polynomials (GCP) are considered in the theory of so-called dessins d’enfant. By definition a complex polynomial \(P\) is called a generalised Chebyshev polynomial if for some \(c_+\) and \(c_-\) the inverse image of the interval \([c_-, c_+]\) is a tree. It is known that the definition is equivalent to a number of purely algebraic conditions. This makes it possible to define generalised Chebyshev polynomials over arbitrary fields and rings.
In this note the author shows: Theorem 1. Let \(P\) be a generalised Chebyshev polynomial, where \(K\) is a field, \(\text{char}\, K = 0\), and \(P\) can be represented as \(P(z) = z^rS(z^k)\). Then either \(r = 0\), or \(k\) is even and \(r = k/2\).
The theorem does not hold in a field \(F\) with \(\text{char}\,F>0\).

MSC:

12E10 Special polynomials in general fields
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A10 Approximation by polynomials
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