For a finite set S = {a1,…, aq
}, consider the polynomial PS
(w) = (w – a1
)(w – a2
) … (w – aq
) and assume that has distinct k zeros. Suppose that PS
(w) is a uniqueness polynomial for entire functions, namely that, for any nonconstant entire functions ɸ and ψ, the equality PS
(ɸ) = cPS
(ψ) implies ɸ = ψ, where c is a nonzero constant which possibly depends on ɸ and ψ. 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) 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.