The homotopy type of spaces of resultants of bounded multiplicity

For positive integers $m,n, d\geq 1$ with $(m,n)\not= (1,1)$ and a field $\Bbb F$ with its algebraic closure $\overline{\Bbb F}$, let $\text{Poly}^{d,m}_n(\Bbb F)$ denote the space of all $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb F [z]$ of monic polynomials of the same degree $d$ such that polynomials $f_1(z),\cdots ,f_m(z)$ have no common root in $\overline{\Bbb F}$ of multiplicity $\geq n$. These spaces were defined by Farb and Wolfson in \cite{FW} as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others in different contexts. In \cite{FW} they obtained algebraic geometrical and arithmetic results about the topology of these spaces. In this paper we investigate the homotopy type of these spaces for the case $\Bbb F =\mathbb{C}$. Our results generalize those of \cite{FW} for $\Bbb F =\Bbb C$ and also results of G. Segal \cite{Se}, V. Vassiliev \cite{Va} and F.Cohen-R.Cohen-B.Mann-R.Milgram \cite{CCMM} for $m\geq 2$ and $n\geq 2$.