  The problem of estimating the multiplicity of the zero of a polynomial when
restricted to the trajectory of a non-singular polynomial vector field, at one
or several points, has been considered by authors in several different fields.
The two best (incomparable) estimates are due to Gabrielov and Nesterenko.
  In this paper we present a refinement of Gabrielov's method which
simultaneously improves these two estimates. Moreover, we give a geometric
description of the multiplicity function in terms certain naturally associated
polar varieties, giving a topological explanation for an asymptotic phenomenon
that was previously obtained by elimination theoretic methods in the works of
Brownawell, Masser and Nesterenko. We also give estimates in terms of Newton
polytopes, strongly generalizing the classical estimates.
