Séminaire Lotharingien de Combinatoire, B37b (1997), 36 pp.

Mohamed El Marraki, Nicolas Hanusse, Jörg Zipperer, Alexander Zvonkin

Cacti, Braids and Complex Polynomials

Abstract. The study of the topological classification of complex polynomials began in the XIX-th century by Luroth, Clebsch and Hurwitz. In the works of Zdravkovska and Khovanskii the problem is reduced to a purely combinatorial one, namely the study of a certain action of the braid groups on a class of tree-like figures that we, following Goulden and Jackson, call "cacti".

Using explicit computation of the braid group orbits, enumerative results of Goulden and Jackson, 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).

Received: January 12, 1997; Accepted: May 29, 1997.

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