47 research outputs found
Multiplicity estimates, analytic cycles and Newton polytopes
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
Multiplicity Estimates: a Morse-theoretic approach
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.Comment: Minor revision; To appear in Duke Math. Journa