The homotopy type of spaces of resultants of 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 (f 1 (z), · · · , f m (z)) ∈ K[z] m of K-coefficients monic polynomials of the same degree d such that polynomials {f k (z)} m k=1 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 [17], and they may be also considered as the real analogues of the spaces studied by B. Farb and J. Wolfson [4]. In this paper, we shall determine their homotopy types explicitly and generalize the previous results obtained in [17] and [11].

The homotopy type of spaces of real resultants with bounded multiplicity

“…We shall now concentrate on the homotopy type of the space Poly d,3 1 (R). Lemma 3.1 ([10], [9]). (i) π 1 (Poly d, 3 1 (R)) = Z for any d ≥ 1.…”

“…Note that the homotopy type of Poly d,m n (F) has been extensively studied for F = C (e.g. [1], [14], [4], [16], [6], [9]). In this article we shall investigate the case F = R. Recall the following already established results.…”

Spaces of non-resultant systems of bounded multiplicity determined by a toric variety

“…In an earlier paper [24] we determined the homotopy type of the space Poly d,m n (F) explicitly for the case F = C and obtained the following homotopy stability result.…”

“…

The space of non-resultant systems of bounded multiplicity for a toric variety X is a generalization of the space of rational curves on it. In our earlier work [24] we proved a homotopy stability theorem and determined explicitly the homotopy type of this space for the case X = CP m . In this paper we consider the case of a general non-singular toric variety and prove a homotopy stability theorem generalising the one for CP m .

…”

“…[24]). Let d, m, n ∈ N be positive integers with (m, n) = (1, 1), and let i d,m n : Poly d,m n (C) → Ω 2 d CP mn−1 ≃ ΩS 2mn−1 denote the natural map given by…”