On differential rings of entire functions

Authors:
A. H. Cayford and E. G. Straus

Journal:
Trans. Amer. Math. Soc. **209** (1975), 283-293

MSC:
Primary 30A98

DOI:
https://doi.org/10.1090/S0002-9947-1975-0382671-1

MathSciNet review:
0382671

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider an entire function $f$ which is a solution of the differential equation \[ [{c_0}(z) + {c_1}(z)D + \ldots + {c_m}(z){D^m}]({f^n}) = P(f,fâ, \ldots ,{f^{(k)}})\] where ${c_i}(z)$ are entire functions in a differential ring $R$ and $P$ is a polynomial in a differential field related to $R$. We prove the following THEOREM. *If $f$ satisfies the equation above then $f$ is of finite type in case $R = {\mathbf {C}}$ and of finite exponential order in case $R = {\mathbf {C}}[z]$*. We use this result to prove a conjecture made in [2] that entire functions of order $\rho < s$, all of whose derivatives at $s$ points are integers in an imaginary quadratic number field, must be solutions of linear differential equations with constant coefficients and therefore of order $\leqslant 1$.

- Enrico Bombieri,
*Algebraic values of meromorphic maps*, Invent. Math.**10**(1970), 267â287. MR**306201**, DOI https://doi.org/10.1007/BF01418775 - Afton H. Cayford,
*A class of integer valued entire functions*, Trans. Amer. Math. Soc.**141**(1969), 415â432. MR**244486**, DOI https://doi.org/10.1090/S0002-9947-1969-0244486-1 - B. Ja. Levin,
*Distribution of zeros of entire functions*, American Mathematical Society, Providence, R.I., 1964. MR**0156975** - E. G. Straus,
*Differential rings of meromorphic functions*, Acta Arith.**21**(1972), 271â284. MR**308418**, DOI https://doi.org/10.4064/aa-21-1-271-284
---,

*Differential rings of analytic functions of a nonarchimedean variable*, Diophantine Approximation and its Applications, Academic Press, New York, 1973, pp. 295-308.

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Keywords:
Integer valued entire function,
linear differential operator,
approximation by algebraic integers,
growth rate

Article copyright:
© Copyright 1975
American Mathematical Society