Zeros of ultrametric meromorphic functions \(f'f^n(f-a)^k-\alpha\). (English) Zbl 1237.30010
Summary: Let \(K\) be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. Similarly to the Hayman problem, here we study meromorphic functions in \(K\) or in an open disk that are of the form \(f'f^n(f-a)^k-\alpha\) with \(\alpha\) a small function, in order to find sufficient conditions on \(n, k\) assuring that they have infinitely many zeros. We first define and characterize a special value for a meromorphic function and check that, if it exists, it is unique. So, such values generalize Picard exceptional values.
MSC:
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
| 30G06 | Non-Archimedean function theory |
| 12J25 | Non-Archimedean valued fields |
References:
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| [2] | DOI: 10.1007/BF02568432 · Zbl 0697.30047 · doi:10.1007/BF02568432 |
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