Nonstandard arithmetic of iterated polynomials

LetK be an algebraic number field of finite degree andf(X,T) a polynomial overK. For eachφ(X)∈ZX], we denote byE(φ) the set of all integersa with φ ^m (a) =φ ⁿ (a) for somem≠n. In this paper, we give a condition for a polynomialφ(X)∈ZX] to satisfy the following; If forn∈N, there existr∈K anda∈Z−E(φ) such thatf r, φ ^m (a)=0, then there exists a rational functiong(X) overK andk∈N such thatf(g(T)), φ ^k (T))=0 .