The homotopy type of spaces of real resultants with bounded multiplicity

TL;DR

This paper determines the explicit homotopy types of spaces of real polynomials with bounded multiplicity of common roots, generalizing previous results and extending understanding of their topological structure.

Contribution

It explicitly computes the homotopy types of spaces of real resultants with bounded multiplicity, extending prior work by Arnold, Vassiliev, Farb, and Wolfson.

Findings

Homotopy types of the spaces are explicitly determined.

Generalization of previous results on polynomial spaces.

Connections to topological and algebraic properties of polynomial spaces.

Abstract

For positive integers $d,m,n\geq 1$ with $(m,n)\not= (1,1)$ and $\Bbb K=\Bbb R$ or $\Bbb C$, let $Q^{d,m}_{n}(\Bbb K)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb K [z]^m$ of $\Bbb K$-coefficients monic polynomials of the same degree $d$ such that polynomials $\{f_k(z)\}_{k=1}^m$ have no common {\it real} root of multiplicity $\geq n$ (but may have complex common root of any multiplicity). %% These spaces can be regarded as one of generalizations of the spaces defined and studied by Arnold and Vassiliev, and they may be also considered as the real analogues of the spaces studied by B. Farb and J. Wolfson. In this paper, we shall determine their homotopy types explicitly and generalize some previously obtained results.