The homotopy type of spaces of real resultants with bounded multiplicity

Abstract

For each pair (m, n) = (1, 1) of positive integers and an arbitrary field F with its algebraic closure F, let Poly d,m n (F) denote the space of m-tuplesThese spaces were first explicitly defined and studied in an algebraic setting by B. Farb and J. Wolfson, in order to prove algebraic analogues of certain topological results of Arnold, Segal, Vassiliev and others. They possess certain stability properties, which have attracted a considerable interest. We have already proved that homotopy stability holds for these spaces and determined their stable homotopy types explicily for the case F = C. We also did the same for the case F = R, under the assumption mn ≥ 4. However, when mn = 3 we had to be satisfied with homological stability. In this paper we show that homotopy stability holds for the space Poly d,m n (R) in the case (m, n) = (3, 1).