ENUMERATION OF ALGEBRAIC AND TROPICAL SINGULAR HYPERSURFACES

We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a delta dimensional linear space of degree d real hyper-surfaces containing 1/delta! (gamma(n)d(n))delta + O(d(n delta-1)) hypersurfaces with delta real nodes, where gamma(n) are positive and given by a recursive formula. This is asymptotically comparable to the number 1/delta! ((n + 1)(d - 1)(n))(delta) + O(d(n()(delta-)(1))) of complex hypersurfaces having delta nodes in a delta dimensional linear space. In the case delta = 1 we give a slightly better leading term.