The homotopy type of spaces of resultants of bounded multiplicity

TL;DR

This paper studies the homotopy type of spaces of tuples of polynomials over the complex numbers with bounded multiplicity conditions, generalizing previous results and connecting algebraic geometry with topology.

Contribution

It extends the understanding of the homotopy types of polynomial spaces with bounded multiplicity to complex fields, generalizing prior work by Farb, Wolfson, Segal, Vassiliev, Cohen, and Mann.

Findings

Determined the homotopy type of these polynomial spaces over a9bba9bba9bb complex numbers.

Generalized previous results for specific cases of m and n.

Connected algebraic geometric properties with topological homotopy classifications.

Abstract

For positive integers $m,n, d\geq 1$ with $(m,n)\not= (1,1)$ and a field $\Bbb F$ with its algebraic closure $\overline{\Bbb F}$, let $\text{Poly}^{d,m}_n(\Bbb F)$ denote the space of all $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb F [z]$ of monic polynomials of the same degree $d$ such that polynomials $f_1(z),\cdots ,f_m(z)$ have no common root in $\overline{\Bbb F}$ of multiplicity $\geq n$. 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 $\Bbb F =\mathbb{C}$. Our results generalize those of \cite{FW} for $\Bbb F =\Bbb C$ and also results of G. Segal \cite{Se}, V. Vassiliev \cite{Va} and F.Cohen-R.Cohen-B.Mann-R.Milgram \cite{CCMM} for $m\geq 2$ and $n\geq 2$.