Homology of Spaces of Non-Resultant Homogeneous Polynomial Systems in $${\mathbb R}^2$$ R 2 and $${\mathbb C}^2$$ C 2

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The resultant variety in the space of systems of homogeneous polynomials of some given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials \({\mathbb R}^2 \rightarrow {\mathbb R}\), and also the rational cohomology rings of spaces of non-resultant systems and non-m-discriminant polynomials in \({\mathbb C}^2\).