AbstractFor a finite setS= {a1,..., aq}, consider the polynomialPS(w) = (w–a1)(w–a2) ... (w–aq) and assume thathas distinctkzeros. Suppose thatPS(w) is a uniqueness polynomial for entire functions, namely that, for any nonconstant entire functionsɸandψ, the equalityPS(ɸ) =cPS(ψ) impliesɸ=ψ, wherecis a nonzero constant which possibly depends onɸandψ. Then, under the conditionq>k+ 2, we prove that, for any given nonconstant entire functiong, there exist at most (2q-2)/(q –k– 2) nonconstant

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... re functionsfwithf*(S) =g*(S), wheref*(S) denotes the pull-back ofSconsidered as a divisor. Moreover, we give some sufficient conditions of uniqueness polynomials for entire functions.