# Multiplicity of zero (higher dimensional analog)

Consider a sistem of n holomorphic equations with n unknowns in a neighborhood of zero. Suppose that a solution in a neighborhood of 0 is a k-dimensional manifold. I want to associate to it some number n_k, its multiplicity, so that solution of any perturbation of the system has not more than n_k k-dimensional strata. Does such multiplicity exist and what is the right definition for it?

-