Abel Equations: Composition Conjectures and the Model Problem

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On the trigonometric moment problem (2011, background): “This problem has been recently study for simple cycles (zi(t)− zj(t)) in [7] and more generally, for polynomial P , in [1].”
A counterexample to the “composition conjecture" (2002, background): “In particular, the truth of the composition conjecture was established under additional assumption that a, b are not critical points of P (z) (see [4]) and under some other additional assumptions (see [2],[3] and [5]).”
Solution of the polynomial moment problem (2007, background): “= i2,1 ≤ i1,i2 ≤ n. Equality (21) is the simplest example of the equality ϕs(z) = 0,1 ≤ s≤ k,and is equivalent to the statement that P(z) and Q(z) have a non-trivial “compositional right factor” (cf. [9], [25], [19], [22], [23]). Lemma 2.3. Let P(z),Q(z) ∈ C[z]. Then the equalities (22) P(z) = P˜(W(”