Towards topological classification of univariate complex polynomials

The study of the topological classification of complex polynomials began in the XIX-th century by Luroth (1871), Clebsch (1873) and Hurwitz (1891). However, very few things are known today about the problem. A vast computer experiment allowed us to achieve a classification of all the polynomials up to degree 11 (until 1996, such a classification was known up to degree 6). These new data change entirely the global view of the problem and permit to formulate some plausible conjectures that may eventually lead to a solution of the problem.