Note on the spaces of real resultants with bounded multiplicity (Geometry, Algebra and Combinatorics in Transformation group theory)

Abstract

For positive integers d, m, ngeq 1 with (m, n)neq(1, 1) and mathbb{K}=mathbb{R} or mathbb{C}, let Q_{n}^{d, m}(mathbb{K}) denote the space of m-tuples (f_{1}(z), f_{m}(z))in mathbb{K}[z]^{m} of mathbb{K}-coeffcients 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 [16]. In this paper, we study the homotopy types of Q_{n}^{d, m}(mathbb{K}) and announce the results obtained in [12].