Multiplicity of a Zero of an Analytic Function on a Trajectory of a Vector Field

The multiplicity μ of a zero of a restriction of an analytic function P in Cn to a trajectory of a vector field ξ with analytic coefficients is equal to the sum of the Euler characteristics of Milnor fibers associated with a deformation of P . When P is a polynomial of degree p and ξ is a vector field with polynomial coefficients of degree q, this allows one to compute μ in purely algebraic terms, and to give an upper bound for μ in terms of n, p, q, single exponential in n and polynomial in p, q. This implies a single exponential in n bound on degree of nonholonomy of a system of polynomial vector fields in Cn. Introduction Let P (x) be a germ at the origin of an analytic function in C, where x = (x1, . . . , xn), and let ξ = ξ1(x)∂/∂x1 + · · ·+ ξn(x)∂/∂xn be a germ at the origin of an analytic vector field. Suppose that ξ(0) 6= 0, and let γ be a trajectory of ξ through the origin. Suppose that P |γ 6≡ 0, and let μ(P |γ) be the multiplicity of a zero of P |γ at the origin. Let ξP = ξ1∂P/∂x1 + · · ·+ ξn∂P/∂xn be derivative of P in the direction of ξ, and let ξP be the kth iteration of this derivative. We show (Theorem 1) that μ(P |γ) is a sum of the Euler characteristics of “Milnor fibers” Xk = {P̃ = ξP̃ = · · · = ξk−1P̃ = 0} associated with a deformation P̃ of P . For a polynomial P of degree p and a vector field ξ with polynomial coefficients of degree q, Xk are (semi-)algebraic sets. This allows one to compute μ(P |γ) in purely algebraic terms (Theorem 3), and to give an upper bound (Theorem 2) for μ(P |γ) in terms of n, p, q, single exponential in n and polynomial in p and q. This estimate improves previous results [9, 1] which were double exponential in n. For a system Ξ = {ξi} of vector fields in R with polynomial coefficients of degree not exceeding q, this implies a single exponential in n and polynomial in q estimate for the degree of nonholonomy of Ξ, i.e., for the minimal order of brackets of ξi necessary to This research was partially supported by NSF Grant # DMS-9704745. Part of this work was done during a visit of the author to the Fields Institute for Research in Mathematical Sciences, Toronto, Canada. The author thanks A. Khovanskii for fruitful discussions. Typeset by AMS-TEX 1