Some numerical computations on smooth real projective hypersurfaces

For a smooth hypersurface defined by a homogeneous polynomial with real coefficients, algorithms are presented to determine the connected components and compute the Euler characteristic for both the real part of the hypersurface and its complement in real projective space. A bipartite graph, which describes the boundary relationships amongst all connected components, is also computed, which, for example, can be used to determine if a connected component of the real hypersurface is one-sided or two-sided. For curves, the nesting structure and dividing type are also considered. Several examples are used to demonstrate the numerical computations.