On uniqueness of meromorphic functions sharing finite sets

We first study conditions for a polynomial P(w) to satisfy the condition that P(f) = cP(g) implies f = g for any nonzero constant c and nonconstant meromorphic functions f and g on c. Next, we give some sufficient conditions for a finite set S to be a uniqueness range set, namely, to satisfy the condition that f(-1)(S) = g(-1)(S) implies f = g for any nonconstant meromorphic functions f and g on c. For a set S, we consider a polynomial P(w) of degree q := #S which vanishes on S. Let P'(w) have distinct k zeros d(1),...,d(k) and assume that k greater than or equal to 4. We show that, if q > 2k+ 12, P(d(l)) not equal P(d(m)) (1 less than or equal to l < m <less than or equal to> k) and P(d(1))+. . .+ P(d(k)) not equal 0, then S is a uniqueness range set and discuss some other related subjects.