Dans les travaux de Zdravkovska et Khovanskii et Zdravkovska le probl`eme est r'eduit `a une 'etude purement combinatoire d'une certaine action du. R'esum'e L"etude de la classification topologique des polynomes complexes a commenc'e au XIX-`eme siecle par Luroth (1871), Clebsch (1873) et Hurwitz (1891). Using explicit computation of the braid group orbits, enumerative results of, and also establishing some combinatorial invariants of the action, we provide the topological classification of polynomials of degree up to 9 (previous results were known up to degree 6).
In the works of Zdravkovska and Khovanskii and Zdravkovska the problem is reduced to a purely combinatorial one, that of the study of a certain action of the braid groups on a class of tree-like figures that we, following, call "cacti". The study of the topological classification of complex polynomials began in the XIX-th century by Luroth (1871), Clebsch (1873) and Hurwitz (1891).
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