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Transcendental entire functions mapping every algebraic number field into itself

Published online by Cambridge University Press:  09 April 2009

A. J. Van Der Poorten
A. J. Van Der Poorten
Affiliation:
School of MathematicsUniversity of New South Wales
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T. Schneider [1] has shown that a transcendental function with a limited rate of growth cannot assume algebraic values at too many algebraic points. It is not clear however whether a transcendental function may assume algebraic values at all algebraic points in an open set in its domain of analyticity. We show, using a method of B. H. Neumann and R. Rado [2] that this can be the case. Indeed we construct transcendental entire functions which, together with all their derivatives, assume, at every point in every algebraic number field, values in that field.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 2 , May 1968 , pp. 192 - 193
Copyright
Copyright © Australian Mathematical Society 1968

References

[1]Schneider, T., Mathemalische Annalen 121, (1949), 131–140.CrossRefGoogle Scholar
[2]Neumann, B. H. and Rado, R., ‘Monotone functions mapping the set of Rational Numbers onto itself’, J. Aust. Math. Soc. 3 (1963). 281–287.CrossRefGoogle Scholar