Spaces of polynomials with real roots of bounded multiplicity (Q1872573)

Let \(P_n^d(\mathbb{K})\) denote the space of monic polynomials of degree d, over a field \(\mathbb{K}\), which have no real root of multiplicity \(\geqslant n\). The author shows that the ``jet map'' is a homotopy equivalence between \(P_3^d(\mathbb{R})\) and \(\Omega S^2\) up to dimension \([d/3]\). The case \(P_3^d(\mathbb{C})\) is also studied and the result in that case is more complicated, asserting a \(\mathbb{Z}/2\) equivariant homotopy equivalence between a certain limit as \(d\rightarrow \infty\) of \(P_3^d(\mathbb{C})\) and \(\Omega S^5\).