15 January 2016 Multiplicity estimates: A Morse-theoretic approach
Gal Binyamini
Duke Math. J. 165(1): 95-128 (15 January 2016). DOI: 10.1215/00127094-3165220


The problem of estimating the multiplicity of the zero of a polynomial when restricted to the trajectory of a nonsingular 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 of 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.


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Gal Binyamini. "Multiplicity estimates: A Morse-theoretic approach." Duke Math. J. 165 (1) 95 - 128, 15 January 2016. https://doi.org/10.1215/00127094-3165220


Received: 20 July 2014; Revised: 12 January 2015; Published: 15 January 2016
First available in Project Euclid: 4 November 2015

zbMATH: 1334.14031
MathSciNet: MR3450743
Digital Object Identifier: 10.1215/00127094-3165220

Primary: 34M99
Secondary: 11J99

Keywords: Bernstein–Kushnirenko–Khovanskii bound , Milnor fibers , multiplicity estimates for vector fields , Polar varieties

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 1 • 15 January 2016
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