Topology of complements of discriminants and resultants

Abstract

In this paper, we classify the homotopy types of spaces of monic polynomials which have no $n$-fold real roots or spaces of $n$-tuples of monic polynomials which have no common real roots, by using the "scanning method"([9]) and Vassiliev's spectral sequence ([15], [16]). In particular, we show that such spaces are finite dimensional models for the infinite dimensional loop space of spheres.