Abstract

Understanding the possible topologies of real projective algebraic hypersurfaces in a given degree and dimension is a key problem in real algebraic geometry, and can be seen as a natural generalization of Hilbert’s 16th problem. There are two complementary approaches to this problem : searching for new constraints, and conversely building examples to show that the configurations which we could not rule out are in fact realizable. I will present a new technique that builds on previous work by O. Viro and I. Itenberg and allows one to effortlessly define families of real projective algebraic hypersurfaces using already-defined families in lower dimensions as building blocks. The asymptotic (in the degree) Betti numbers of the real parts of the resulting families can then be recovered from the asymptotic Betti numbers of the real parts of the building blocks. Using this technique, I will explain how families of real algebraic hypersurfaces whose real parts have asymptotically large Betti numbers can be constructed in any dimension. The results presented in this talk can also be found in https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/topo.12251.