Multiplicity of a Noetherian Intersection

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 fi = 0 where fi 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. Introduction 1. The main result. A ring K of analytic functions in an open domain U ⊂ C is called a ring of Noetherian functions in U if 1) K contains the ring C[x1, . . . , xn] of polynomials and is finitely generated over that ring; 2) K is closed under differentiation. In other words, for each f ∈ K, all partial derivatives ∂f/∂xi belong to K. In particular, K is a Noetherian ring, which is the origin of the notation “Noetherian function” introduced by Tougeron [T]. A set of m functions ψ = {ψ1, . . . , ψm} is called a Noetherian chain of order m if these functions generate K over C[x1, . . . , xn]. A function φ in K is called a Noetherian function of degree β relative to a Noetherian chain ψ if there exists a polynomial P of degree not exceeding β in n + m variables such that φ = P (x, ψ(x)). The Noetherian chain ψ has degree not exceeding α if each partial derivative ∂ψi/∂xj is a Noetherian function of degree α relative to ψ. Standard Noetherian arguments allow one to prove that the multiplicity of an isolated intersection of Noetherian functions is bounded by a certain function of discrete parameters n, m, α, and β. As usual, these arguments do not provide any effective method of computation of such function. This computation is done in the present paper. The main result of this paper is the following Partially supported by NSF Grant # DMS-9704745 and by NSERC Grant # OGP0156833. Part of this work was done when both authors participated in the program on Singularity Theory and Geometry at the Fields Institute for Research in Mathematical Sciences, Toronto, Canada. Typeset by AMS-TEX 1 2 ANDREI GABRIELOV AND ASKOLD KHOVANSKII Theorem 1. Let φ1, . . . , φn belong to a ring K of Noetherian in U ⊂ C. Suppose that all φi have degree not exceeding β relative to a common Noetherian chain ψ of order m and degree α ≥ 1. Then the multiplicity of any isolated solution of the system of equations φ1 = · · · = φn = 0 does not exceed maximum of the following two numbers: 1 2 Q ( (m + 1)(α− 1)[2α(n + m + 2)− 2m− 2] + 2α(n + 2)− 2)2(m+n), 1 2 Q ( 2(Q + n)(β + Q(α− 1)))2(m+n), where Q = en ( e(n + m) √ n )ln n+1 ( n e2 )n . 2. Pfaff systems with polynomial coefficients. Rings of Noetherian functions and Noetherian chains can be defined in terms of systems of Pfaff equations. This approach is more geometric, and will be used in the proof of the main result. Definition 1. An analytic n-dimensional distribution in an open domain U ⊂ C is defined by