The homotopy type of spaces of resultants of bounded multiplicity

For integers m,n,d≥1 with (m,n)≠(1,1) and a field F with algebraic closure F‾, let Polyd,mn(F) denote the space of all m-tuples (f1(z),…,fm(z))∈F[z]m of monic polynomials of the same degree d such that polynomials f1(z),…,fm(z) have no common root in F‾ of multiplicity ≥n. These spaces were defined by Farb and Wolfson [12] as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others in different contexts. They obtained algebraic geometrical and arithmetic results about them. In this paper we investigate the homotopy type of these spaces for the case F=C. Our results generalize those of [12] for F=C and also results of G. Segal [28], V. Vassiliev [30] and F. Cohen-R. Cohen-B. Mann-R. Milgram [5] for m≥2 and n≥2.