In his study of the geometry of (germs of) zero sets for holomorphic functions of several complex variables, Weierstrass proved around 1880 his famous Preparation Theorem, which reduces this study to parametrized families of complex polynomials with holomorphic coefficients. By adapting these concepts to the continuous setting, one is led to an interesting class of finite covering maps, introduced in and under the name polynomial covering maps. From an algebraic point of view, polynomial covering maps correspond to conjugacy classes of homomorphisms of fundamental groups into the Artin braid groups. The theory of this special class of finite covering maps is thereby closely related to the theory of braid groups.

The first section of Chapter III contains the basic definitions of Weierstrass polynomials and the covering maps associated with them. The exploration of these concepts will be our main concern in Chapters III and IV.

In Chapter III we present three characterizations of the class of polynomial covering maps. First we prove the existence of a canonical n–fold polynomial covering map onto the complement Bn = ℂn\Δ of the discriminant set Δ in complex n–space ℂn, from which every n–fold polynomial covering map is induced by a pull–back construction. The second characterization is an embedding criterion, according to which a finite covering map onto a space X is equivalent to a polynomial covering map if and only if it embeds (fibrewise) into the trivial complex line bundle over X. The third characterization describes the n–fold polynomial covering maps onto X in terms of conjugacy classes of homomorphisms of the fundamental group of X into the Artin braid group B(n) on n strings.