2015/01/14 by Gal Binyamini, Binyamini, Gal
Computer Science · Mathematics · #14C17 (secondary) #34C08 (primary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14C17 #msc:34C08
paper · pdf · doi:10.48550/arxiv.1501.03247
Minor revision; To appear in Selecta Mathematica
openalex publication_date 2015/01/14 · arxiv created 2015/08/13 · arxiv updated 2015/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of estimating the intersection multiplicity between an algebraic variety and a Pfaffian foliation, at every point of the variety. We show that this multiplicity can be majorized at every point p by the local algebraic multiplicity at p of a suitably constructed algebraic cycle. The construction is based on Gabrielov's complex analog of the Rolle-Khovanskii lemma. We illustrate the main result by deriving similar uniform estimates for the complexity of the Milnor fiber of a deformation (under a smoothness assumption) and for the order of contact between an algebraic hypersurface and an arbitrary non-singular one-dimensional foliation. We also use the main result to give an alternative geometric proof for a classical multiplicity estimate in the context of commutative group varieties.