Séminaire Lotharingien de Combinatoire, B37b (1997), 36
Mohamed El Marraki, Nicolas Hanusse, Jörg Zipperer, Alexander Zvonkin
Cacti, Braids and Complex Polynomials
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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