Relationship between polynomials with multiple roots and rational functions with common roots - Kamiyama, Yasuhiko (MATHEMATICA SCANDINAVICA, 2005)
For F= R or C, let ¶^l_k,n(F)l denote the space of monic polynomials f(z) over F of degree k and such that the number of n-fold roots of f(z) is at most l. Let X^l_k, n(F) denote the space consisting of all n-tuples (p_1(z),…,p_n(z)) of monic polynomials over F of degree k and such that there are at most l roots common to all p_i (z). In this paper, we prove that P^l_k,n(F) and X_[kn],n^l(F) are stably homotopy equivalent. In fact, they are homotopy equivalent when F = C and (n, l) ot= (2, 0). We also consider the case that n-fold roots and common roots are not real. These results generalize previous results concerning these spaces.