Spaces of polynomials without 3-fold real roots

Let $P_{n}^{d}(\mathbb{R})$ denote the space consisiting of all monic polynomials $f(z) \in \mathbb{R}[z]$ of degree $d$ which have no real roots of multplicity $\geq n$. In this paper we study the homotopy types of the spaces $P_{n}^{d}(\mathbb{R})$ for the case $n = 3$.