Dimension of an analytic set

Summary

The dimension of a complex analytic variety at a point measures how many complex parameters are needed to describe it near that point. At a regular point it is the dimension of the set as a complex manifold. At a singular point it is the largest dimension of the manifold pieces nearby. It agrees with the Krull dimension of the local ring of the set, and with the number $d$ of the local parametrization theorem.