On the existence of unique range sets generated by non-critically injective polynomials and related issues

Abstract

In this paper, we prove the existence of non-critically injective polynomials whose set of zeros form unique range sets that answers one of the most awaited and fundamental questions of uniqueness theory of entire and meromorphic functions. We also show that there exist some unique range sets and their generating polynomials which can not be characterized by any of the existing generalized results of unique range sets but as an application of our main theorems the same can be characterized. Moreover, as an application of our main results we prove that the cardinality of a unique range set does not always depend upon the number of distinct critical points of its generating polynomial.