The homotopy type of spaces of resultants of bounded multiplicity

Abstract

For positive integers $m,n,d\ge 1$ with $(m,n)\ne (1,1)$ and a field $\mathbb{F}$ with its algebraic closure $\overline{\mathbb{F}}$, let ${\text{Poly}}_n^{d,m}(\mathbb{F})$ denote the space of all $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \mathbb{F}[z]$ of monic polynomials of the same degree $d$ such that polynomials $f_1(z),\cdots ,f_m(z)$ have no common root in $\overline{\mathbb{F}}$ of multiplicity $\ge n$. These spaces were defined by Farb and Wolfson in \cite{FW} as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others in different contexts. In \cite{FW} they obtained algebraic geometrical and arithmetic results about the topology of these spaces. In this paper we investigate the homotopy type of these spaces for the case $\mathbb{F}=\mathbb{C}$. Our results generalize those of \cite{FW} for $\mathbb{F}=\mathbb{C}$ and also results of G. Segal \cite{Se}, V. Vassiliev \cite{Va} and F.Cohen-R.Cohen-B.Mann-R.Milgram \cite{CCMM} for $m\ge 2$ and $n\ge 2$.