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2007 Finiteness of entire functions sharing a finite set
Hirotaka Fujimoto
Nagoya Math. J. 185: 111-122 (2007).

Abstract

For a finite set $S = \{a_1, \ldots, a_q\}$, consider the polynomial $P_S(w) = (w - a_1)(w - a_2) \cdots (w - a_q)$ and assume that $P'_S(w)$ has distinct $k$ zeros. Suppose that $P_S(w)$ is a uniqueness polynomial for entire functions, namely that, for any nonconstant entire functions $\phi$ and $\psi$, the equality $P_S(\phi) = c P_S(\psi)$ implies $\phi = \psi$, where $c$ is a nonzero constant which possibly depends on $\phi$ and $\psi$. Then, under the condition $q > k + 2$, we prove that, for any given nonconstant entire function $g$, there exist at most $(2q\!-\!2)/(q-k-2)$ \linebreak nonconstant entire functions $f$ with $f^*(S) = g^*(S)$, where $f^*(S)$ denotes the pull-back of $S$ considered as a divisor. Moreover, we give some sufficient conditions of uniqueness polynomials for entire functions.

Citation

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Hirotaka Fujimoto. "Finiteness of entire functions sharing a finite set." Nagoya Math. J. 185 111 - 122, 2007.

Information

Published: 2007
First available in Project Euclid: 23 March 2007

zbMATH: 1127.30014
MathSciNet: MR2301460

Subjects:
Primary: 30D35

Rights: Copyright © 2007 Editorial Board, Nagoya Mathematical Journal

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