On the equation P(f) = Q(g), where P, Q are polynomials and f, g are entire functions

In 1922 Ritt described polynomial solutions of the functional equation $P(f) = Q(g).$ In this paper we describe solutions of the equation above in the case when $P, Q$ are polynomials while $f, g$ are allowed to be arbitrary entire functions. In fact, we describe solutions of the more general functional equation $s = P(f) = Q(g),$ where $s, f, g$ are entire functions and $P, Q$ are arbitrary rational functions. As an application we solve the problem of description of "strong uniqueness polynomials" for entire functions.