Intersection theory and Hilbert function

Khovanskiĭ, Askold

doi:10.1007/s10688-011-0033-6

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2014

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“…The proof is analogous to that of [2] (suggested by Kollar). See also [17] for a different approach.…”

Section: Toric Classesmentioning

confidence: 99%

Multiplicity estimates, analytic cycles and Newton polytopes

Binyamini

2014

Preprint

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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 successfully applied to several problems in transcendental number theory.We propose an alternative approach to this problem using algebraic cycles and their local analytic structure. 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.

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“…The proof is analogous to that of [2] (suggested by Kollar). See also [17] for a different approach.…”

Section: Toric Classesmentioning

confidence: 99%