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 \xi = \xi_1(x) d/dx_1 + ... + \xi_n(x) d/dx_n be a germ at the origin of an analytic vector field. Suppose that \xi(0) != 0, and let \gamma be a trajectory of \xi through the origin. Suppose that P|_\gamma /\equiv 0, and let \mu(P|_\gamma) be the multiplicity of a zero of P|_\gamma at the origin. Let \xi P = \xi_1 dP/dx_1 + ... + \xi_n dP/dx_n be derivative of P in the direction of \xi, and let \xi^kP be the kth iteration of this derivative. We give a formula (Theorem 1) for \mu(P|_\gamma) in terms of the Euler characteristic of the Milnor fibers defined by a deformation of P, \xi P, ..., \xi^{n-1}P . For a polynomial P of degree p and a vector field \xi with polynomial coefficients of degree q, this allows one to compute \mu(P|_\gamma) in purely algebraic terms (Theorem 2), and to give an estimate (Theorem 3) for \mu(P|_\gamma) 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.