Urs, Ursim, and Non-Urs for p-adic Functions and Polynomials Original Research Article

LetWbe an algebraically closed field of characteristic zero, and letKbe an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. LetQnbe the polynomialxn−xn−1+kfor any constantk≠0, (n−1)n−1/nn. LetTn(k) be the set ofndistinct zeros ofQn. For everyngreater-or-equal, slanted9, we show thatTn(k) is ann-point unique range set (ignoring multiplicities) for bothW[x] and the set image(K) of entire functions inK. However, for everyn>0, we also show thatTn(k) is not a unique range set (counting or ignoring multiplicities) forW(x) and therefore, is also not a unique range set for the set of p-adic meromorphic functions (this was also separately found by Chung-Chun Yang and Xin-Hou Hua). In the same way, we show that there exist no bi-urs for p-adic meromorphic functions of the form ({a, b, c}, {ω}). Moreover, for everyngreater-or-equal, slanted5, we show that the only linear fractional functions preserving a setTn(k) is the identity, something which was asked (in particular) in Boutabaa and Escassut, “On Uniqueness of p-adic Meromorphic Functions,”Proc. Amer. Math. Soc.(1988).