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When is the iterate of a formal power series odd?

Published online by Cambridge University Press:  09 April 2009

Bruce Reznick
Bruce Reznick
Affiliation:
Department of Mathematics University of CaliforniaBerkeley, California 94720, U.S.A.
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Abstract

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The formal power series (fps) f(z) = Σ∞i=1ai zi is homozygous mod k if ai ≠ 0 implies i ≡ j mod k. This generalizes even and odd fps. If f is homozygous mod k then all iterates of f (fn = f ο fn−1) are also homozygous mod k, but the converse is false–there are many non-odd fps f for which f(f(z)) = z. It is shown that if f is not homozygous mod k but fn is homozygous, then fnr(z) = z for some r. If all coefficienrs ar real then, in fact, f(f(z)) = z.

Subject classification (Amer. Math Soc. (MOS) 1970): 39 A 05.

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 1 , August 1979 , pp. 62 - 66
Copyright
Copyright © Australian Mathematical Society 1979

References

Baker, I. N. (1962), ‘Permutable power series and regular iteration’, J. Austral. Math. Soc. 2, 265–294.CrossRefGoogle Scholar