Urs, ursim, and non-urs for p-adic functions and polynomials

Let W be an algebraically closed field of characteristic zero, and let K be an algebraically closed field of characteristic zero, complete for an ultr ametric absolute value. Let Q(n) be the polynomial x(n) - x(n-1) + k for an y constant k not equal 0, (n - 1)(n-1)/n(n). Let T-n(k) be the set of n dis tinct zeros of Q(n). For every n greater than or equal to 9, we show that T -n(k) is an M-point unique range set (ignoring multiplicities) for both W[x ] and the set A(K) of entire functions in K. However, for every n > 0, we a lso show that T-n(k) is not a unique range set (counting or ignoring multip licities) for W(x) and therefore, is also not a unique range set for the se t 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}; {omega}). Mor eover, for every n greater than or equal to 5. we show that the only linear fractional Functions preserving a set T-n(k) is the identity, something wh ich was asked (in particular) in Boutabaa and Escassut, "On Uniqueness of p -adic Meromorphic Functions," Proc. Amer. Math. Soc. (1988). (C) 1999 Acade mic Press.