Hans-Christian Graf v. Bothmer
Hannover
Counting Components Heuristically
Via the Weil-Conjectures one can obtain a lot of geometric information
about an algebraic variety X defined over ZZ by counting points over a
finite field F_p. Unfortunatedly in many interesting examples it is
impossible to count all point because there simply are to many of them.
By evaluating the equations of X in a number of random points one can
estimate the total number of points with sufficient precision to obtain
information about the number of components of X and the codimensions of
these components. This method is fast and easily parallelizable, but
yields results that are only "probably right". In this talk I will
explain this method in more detail and show how it can be applied to
the construction of rational surfaces in IP^4 and to the Poincar\'e
center problem.