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Appendix B Chebyshev Polynomials and Their Inverses

These polynomials were discovered by Pafnuty Chebyshev (1821–1894) when he was considering the problem of the best approximation of a given function by polynomials of degree n. They play an important role in approximation theory. Rather surprising is the fact that these polynomials became useful in algebra: the problem from which they originally appeared is far from algebra, and even their definition uses transcendental functions. Nevertheless, in some algebraic problems, the series Tn of Chebyshev polynomials appears along with the polynomials P.x/ D x. From a “philosophical” point of view, these two classes result from the existence of two families of finite groups of projective transformations of the space P: cyclic groups Cn and dihedral groupsDn. In complex analysis, the class of polynomials x extends to the family of multivalued analytic functions x , ̨ 2 , which contains, along with the polynomials x, their inverses x and satisfies the composition relation .x/ D x . In a similar manner, we extend the class of Chebyshev polynomials Tn to the family of multivalued analytic functions T ̨ , ̨ 2 , which contains, along with the polynomials Tn, their inverses T1=n and satisfies the composition relation Tˇ ı T ̨ DT ̨ˇ . A multivalued function can be defined without the notion of analytic continuation, just by giving its set of values at each point. This sometimes helps us in carrying over the definition of the multivalued function to an arbitrary field (where the operation of analytic continuation is not defined). For example, for positive