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

TL;DR

This paper studies the topology of polynomial spaces over the real numbers where polynomials have no common roots of high multiplicity, extending previous complex case results to the real case.

Contribution

It extends the explicit homotopy type analysis of these polynomial spaces from complex to real coefficients, providing new topological insights.

Findings

Explicit homotopy type for real coefficient spaces

Comparison with complex case results

New topological properties identified for real polynomial spaces

Abstract

For each pair $(m,n)$ of positive integers with $(m,n)\not= (1,1)$ and an arbitrary field $\bf F$ with algebraic closure $\overline{\bf F}$, let $\rm Po^{d,m}_n(\bf F)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \bf F [z]^m$ of $\bf F$-coefficients monic polynomials of the same degree $d$ such that the polynomials $\{f_k(z)\}_{k=1}^m$ have no common root in $\overline{\bf F}$ of multiplicity $\geq n$. These spaces $\rm Po^{d,m}_n(\bf F)$ 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 $\bf F =\Bbb C$. In this paper, we investigate the case $\bf F =\Bbb R$.