The homotopy type of spaces of real resultants with bounded multiplicity

Abstract

For positive integers $d, m, n \geq 1$ with $(m, n) \neq (1, 1)$ and $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, let $\mathbb{Q}^{d,m}_{n}(\mathbb{K})$ denote the space of $m$-tuples $(f_{1}(z), \ldots, f_m(z)) \in \mathbb{K} [z]^{m}$ of $\mathbb{K}$-coefficients monic polynomials of the same degree $d$ such that polynomials $\{f_{k}(z)\}_{k=1}^{m}$ have no common 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 Farb–Wolfson. In this paper, we shall determine their homotopy types explicitly and generalize our previous results.