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For any natural $d \ge k \ge 2$ we calculate the cohomology groups of thespace of homogeneous polynomials $R^2 \to R$ of degree $d$, which do not vanishwith multiplicity $\ge k$ on real lines. For $k=2$ this problem provides thesimplest example of the situation, when the "finite degree" invariants ofnonsingular objects are not a complete system of invariants.

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