On the space of real polynomials without multiple critical values (Q1316810)

Let \(L_ \mu\) be the variety of the polynomials of the form \[ z^{\mu+1} + a_{\mu-1} z^{\mu-1} + a_{\mu-2} z^{\mu-2} + \cdots + a_ 0,\quad a_ i \in \mathbb{R}, \] regarded as functions from \(\mathbb{C}\) to \(\mathbb{C}\) and having exactly \(\mu\) different critical values. The author computes the group \(H_ 0 (L_ \mu)\). The results are generalized to the space of rational functions. He also formulates a conjecture concerning the homotopy type of the connected components of the variety \(L_ \mu\).