We consider the problem of estimating the multiplicity of a polynomial when
restricted to the smooth analytic trajectory of a (possibly singular)
polynomial vector field at a given point or points, under an assumption known
as the D-property. Nesterenko has developed an elimination theoretic approach
to this problem which has been widely used in transcendental number theory.
We propose an alternative approach to this problem based on more local
analytic considerations. In particular we obtain simpler proofs to many of the
best known estimates, and give more general formulations in terms of Newton
polytopes, analogous to the Bernstein-Kushnirenko theorem. We also improve the
estimate's dependence on the ambient dimension from doubly-exponential to an
essentially optimal single-exponential.Comment: Some editorial modifications to improve readability; No essential
mathematical change