Relationship between polynomials with multiple roots and rational functions with common roots

Abstract

For F = R or C, let P l k,n (F ) 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(F ) are stably homotopy equivalent. In fact, they are homotopy equivalent when F = C and (n, l) = (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.

The homology of spaces of polynomials with roots of bounded multiplicity

“…ξ l (S 2n−2 ). Geometry & Topology Monographs 13 (2008)The homology of spaces of polynomials 285 Kamiyama[12]) For l ≥ 1 and n ≥ 2, there is a homotopy equivalence…”