Families of commuting formal power series, semicanonical forms and iterative roots
Abstract
We consider for a given formal power series F(x) = ρx + c2x2 + ... , ρ ≠ 0, with complex coefficients and for a given integer N > 1 the functional equation GN = F where G is again a formal series (G is called an iterative root of F). If ρ is a root of 1 and F is not conjugate to its linear part we derive a criterion for the existence of solutions G and describe the general solution. Representations of the coefficients of G by means of universal polynomials are given, also in the case where ρ is not a root of 1 (where existence is almost trivial). Our main tools are maximal families of commuting series (i.e. the Aczél-Jabotinsky equation of third type) and semicanonical forms.
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Institut für Mathematik, Universität Graz, Austria Austria
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