  For positive integers $d,m,n\geq 1$ with $(m,n)\not= (1,1)$ and $\Bbb K=\Bbb
R$ or $\Bbb C$, let $Q^{d,m}_{n}(\Bbb K)$ denote the space of $m$-tuples
$(f_1(z),\cdots ,f_m(z))\in \Bbb K [z]^m$ of $\Bbb K$-coefficients monic
polynomials of the same degree $d$ such that polynomials $\{f_k(z)\}_{k=1}^m$
have no common {\it real} root of multiplicity $\geq n$ (but may have complex
common root of any multiplicity). %% These spaces can be regarded as one of
generalizations of the spaces defined and studied by Arnold and Vassiliev, and
they may be also considered as the real analogues of the spaces studied by B.
Farb and J. Wolfson. In this paper, we shall determine their homotopy types
explicitly and generalize some previously obtained results.
