Spaces of non-resultant systems of bounded multiplicity with real coefficients

Abstract

For each pair $(m,n)$ of positive integers with $(m,n)\ne (1,1)$ and an arbitrary field $\mathbf{F}$ with algebraic closure $\overline{\mathbf{F}}$, let $\mathrm{P}{\mathrm{o}}_{\mathrm{n}}^{\mathrm{d},\mathrm{m}}(\mathbf{F})$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \mathbf{F}[\mathbf{z}{]}^{\mathbf{m}}$ of $\mathbf{F}$-coefficients monic polynomials of the same degree $d$ such that the polynomials $\{f_k(z){\}}_{k=1}^m$ have no common root in $\overline{\mathbf{F}}$ of multiplicity $\ge n$. These spaces $\mathrm{P}{\mathrm{o}}_{\mathrm{n}}^{\mathrm{d},\mathrm{m}}(\mathbf{F})$ were first defined and studied by B. Farb and J. Wolfson as generalizations of spaces first studied by Arnold, Vassiliev and Segal and others in several different contexts. In previous we determined explicitly the homotopy type of this space in the case $\mathbf{F}=\mathbb{C}$. In this paper, we investigate the case $\mathbf{F}=\mathbb{R}$.