Abstract
We consider the affine real variety \(V=F^{-1}(0)\) in \(\mathbf{R}^n\) where \(F\) is given by two quadratic forms. We announce several recent results, obtained in collaboration with various coauthors, about the topology of the generic \(V\) and of its deformations \(V_t=F^{-1}(t)\), especially those that are smooth. This work started many years ago [16] with the case where the two quadratic forms are simultaneously diagonalizable and with some restrictions. Now we have the result without restrictions and including the non-diagonalizable case so we have a complete topological description of \(V\) in all generic cases. We have also results about the topology of the intersection of \(V\) with a half-space \(\{x_i\ge 0\}\) and the topology of the various deformations \(V_t\). However, in these cases the results are for the moment not as complete as those obtained for the topology of \(V\). Proofs are referred to published articles or only outlined when complete proofs are still to appear.
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