Homology of spaces of non-resultant polynomial systems in R^2 and C^2

The resultant veriety in the space of systems of homogeneous polynomials of 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 $R^2\to R$, and also the rational cohomology groups of similar systems in $C^2$.