For positive integers $d,m,n$ with $(m,n)\not= (1,1)$ let
$Poly^{d,m}_n$ denote the space of $m$-tuples
$(f_1(z),\cdots ,f_m(z))$ of complex monic polynomials of the same
degree $d$ such that they have no common root of multiplicity $\geq n$.
When $m=1$ or $n=1$, the homotopy type of it was already well studied.
In this talk we study its the homotopy type for $m>1$ and $n>1$
and try to consider the generalization of the results due to G. Segal
and V. Vassiliev.
This talk is based on the joint work with A. Kozlowski (University of
Warsaw).