Multiplicity of a Noetherian Intersection

Andrei Gabrielov and Askold Khovanskii

October 28, 1997

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.
In this paper, we give an effective bound on 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.
In the one-dimensional case, such an estimate is known and has
applications in number theory and control theory.
Multi-dimensional case presented in this paper provides a solution
of a rather old problem concerning finiteness properties of
transcendental functions defined by algebraic
partial differential equations.