Families of commuting formal power series, semicanonical forms and iterative roots

Ludwig Reich

Published: 1994-09-30

Vol. 8 (1994)

Families of commuting formal power series, semicanonical forms and iterative roots

Ludwig Reich

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 G^N = 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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Reich, L. (1994). Families of commuting formal power series, semicanonical forms and iterative roots. Annales Mathematicae Silesianae, 8, 189=201. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14234

References

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Vol. 8 (1994)
Published: 1994-09-30

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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