Counting complex points of surfaces in ℂ 2

We deal with complex points of two-dimensional surfaces. A short survey of basic results about complex points of smooth surfaces in ℂ 2 is presented at the beginning. For algebraic surfaces, a formula is proven which expresses the number of complex points as the local degree of an explicitly constructed polynomial endomorphism. Using this formula, some estimates for the number of complex points and the Maslov index are obtained in terms of algebraic degrees of defining equations with special attention paid to graphs of planar endomorphisms. Some estimates for the expected number of complex points of a random planar endomorphism are also obtained. © 2006 Springer Science+Business Media, Inc.