A differential ring of analytic functions in several complex variables is called a ring of Noetherian functions if it is finitely generated as a ring and contains the ring of all polynomials. The multiplicity of an isolated solution of a system of $n$ equations $f_i=0$, where $f_i$ belong to a ring of Noetherian functions in $n$ complex variables, can be expressed in terms of the Euler characteristics of the generalized Milnor fibers associated with this system. This provides an effective upper bound on this multiplicity. In combination with constructive resolution of singularities over the fields of characteristic zero, this allows one to obtain an effective upper bound on the complexity of the resolution of singularities defined by Noetherian functions. For $n=1$, Noetherian functions are soultions of a system of algebraic ordinary differential equations. The upper bound on their multiplicity implies an effective upper bound for degree of nonholonomy of a system of algebraic vector fields, an important problem in control theory.