Enumeration of singular hypersurfaces via tropical geometry

We discuss a tropical approach to enumeration of
singular complex and real hypersurfaces in toric varieties. In particular,
we establish a correspondence theorem between singular tropical and
algebraic hypersurfaces and demonstrate a multi-dimensional version of
a lattice path algorithm. As application we obtain a lower bound to the
maximal possible number of real singular hypersurfaces in a generic real pencil.