Homology of spaces of homogeneous polynomials in $R^2$ without multiple zeros

For any natural $d\ge k\ge 2$ we calculate the cohomology groups of the space of homogeneous polynomials $R^2\to R$ of degree $d$, which do not vanish with multiplicity $\ge k$ on real lines. For $k=2$ this problem provides the simplest example of the situation, when the “finite degree” invariants of nonsingular objects are not a complete system of invariants.