Multiplicity of a zero of an analytic function on a trajectory of a vector field

Abstract

Let P(x) be a germ at the origin of an analytic function in C^n, where x = (x_1,..., x_n), and let ξ= ξ_1(x) d/dx_1 + ... + ξ_n(x) d/dx_n be a germ at the origin of an analytic vector field. Suppose that ξ(0) != 0, and let γbe a trajectory of ξthrough the origin. Suppose that P|_γ/\equiv 0, and let μ(P|_γ) be the multiplicity of a zero of P|_γat the origin. Let ξP = ξ_1 dP/dx_1 + ... + ξ_n dP/dx_n be derivative of P in the direction of ξ, and let ξ^kP be the kth iteration of this derivative. We give a formula (Theorem 1) for μ(P|_γ) in terms of the Euler characteristic of the Milnor fibers defined by a deformation of P, ξP, ..., ξ^{n-1}P . For a polynomial P of degree p and a vector field ξwith polynomial coefficients of degree q, this allows one to compute μ(P|_γ) in purely algebraic terms (Theorem 2), and to give an estimate (Theorem 3) 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 which were doubly exponential in n.