A uniqueness polynomial generating a unique range set and vice versa (Q1930680)

In this paper, the authors establish a kind of equivalence of different notions of unique range sets and uniqueness polynomials. They find a polynomial (of course not necessarily critically injective) which can bring all variants of unique range sets and uniqueness polynomials under a single umbrella. Moreover, the authors give an example which shows that the hypothesis of Theorem 1.3 does not ensure a polynomial to be a SUPM; rather, in addition, the condition stated in (vi) is essential.