Unique Range Sets for Meromorphic Functions Constructed without an Injectivity Hypothesis

Abstract

A set is called a unique range set (counting multiplicities) for a particular family of functions if the inverse image of the set counting multiplicities uniquely determines the function in the family. So far, almost all constructions of unique range sets for meromorphic functions are zero sets of polynomials which satisfy an injectivity condition introduced by Fujimoto. A polynomial $P(z)$ satisfies the injectivity condition if $P$ is injective on the zeros of its derivative. In this paper, we will construct examples of unique range sets for meromorphic functions without assuming an injectivity condition.