The homotopy type of spaces of real resultants with bounded multiplicity

Abstract

For positive integers 𝑑, 𝑚, 𝑛 ≥ 1 with (𝑚, 𝑛) ≠ (1, 1) and 𝕂 = ℝ or ℂ, let ℚ^𝑑,𝑚_𝑛(𝕂) denote the space of 𝑚-tuples (𝑓₁(𝑧), …, 𝑓_𝑚(𝑧)) ∈ 𝕂 [𝑧]^𝑚 of 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 ≥ 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.