Finiteness of entire functions sharing a finite set

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.