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

Abstract

For $F= R$ or $C$, let $\P^l_{k,n}(F){l}$ 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 $(p_1(z),\ldots,p_n(z))$ 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 $X_{[\frac{k}{n}],n}^l(F)$ are stably homotopy equivalent. In fact, they are homotopy equivalent when $F = C$ and $(n, l) \not= (2, 0)$. We also consider the case that $n$-fold roots and common roots are not real. These results generalize previous results concerning these spaces.