Some arithmetic properties of the values of entire functions of finite order and their first derivatives

We describe a class of entire functions of finite order which, together with their first derivative, take sufficiently many algebraic values (with certain restrictions on the growth of the degree and height of these values). We show that, under certain conditions, any such function is a rational fun...

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Bibliographic Details
Published in:Sbornik. Mathematics 2019-12, Vol.210 (12), p.1788-1802
Main Author: Yanchenko, A. Ya
Format: Article
Language:English
Subjects:
Algebra
algebraic values
derivative
entire function of finite order
Entire functions
exponentials
Mathematical analysis
Rational functions
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Summary:We describe a class of entire functions of finite order which, together with their first derivative, take sufficiently many algebraic values (with certain restrictions on the growth of the degree and height of these values). We show that, under certain conditions, any such function is a rational function of special form of an exponential. For entire functions of finite order which are not representable in the form of a finite linear combination of exponentials, we obtain an estimate for the number of points (in any fixed disc) at which the values of the function itself and its first derivative are algebraic numbers of bounded degree and height. Bibliography: 8 titles.
ISSN:1064-5616
1468-4802
DOI:10.1070/SM9145