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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 Farband Wolfson in \cite{FW} as generalizations of spaces first studied by Arnold,Vassiliev, Segal and others in different contexts. In \cite{FW} they obtainedalgebraic geometrical and arithmetic results about the topology of thesespaces. In this paper we investigate the homotopy type of these spaces for thecase $\Bbb F =\mathbb{C}$. Our results generalize those of \cite{FW} for $\BbbF =\Bbb C$ and also results of G. Segal \cite{Se}, V. Vassiliev \cite{Va} andF.Cohen-R.Cohen-B.Mann-R.Milgram \cite{CCMM} for $m\geq 2$ and $n\geq 2$.

The homotopy type of spaces of resultants of bounded multiplicity