Inductive solution of the tangential center problem on zero-cycles

Given a polynomial $f\in\mathbb C[z]$ of degree $m$, let $z_1(t),\dots,z_m(t)$ denote all algebraic functions defined by $f(z_k(t))=t$. Given integers $n_1,\dots,n_m$ such that $n_1+\dots+n_m=0$, the tangential center problem on zero-cycles asks to find all polynomials $g\in\mathbb C[z]$ such that $n_1g(z_1(t))+\dots+n_mg(z_m(t))\equiv0$. The classical Center-Focus Problem, or rather its tangential version in important non-trivial planar systems lead to the above problem.
The tangential center problem on zero-cycles was recently solved in a preprint by Gavrilov and Pakovich.
Here we give an alternative solution based on induction on the number of composition factors of $f$ under a generic hypothesis on $f$. First we show the uniqueness of decompositions $f=f_1\circ\dots\circ f_d$ such that every $f_k$ is $2$-transitive, monomial or a Chebyshev polynomial under the assumption that in the above composition there is no merging of critical values.
Under this assumption, we give a complete (inductive) solution of the tangential center problem on zero-cycles. The inductive solution is obtained through three mechanisms: composition, primality and vanishing of the Newton-Girard component on projected cycles.