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Polynomial Partitioning on Varieties of Codimension Two and Point-Hypersurface Incidences in Four Dimensions

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Abstract

We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matoušek, Sharir and Safernová. We also present a bound for the number of incidences between points and hypersurfaces in the four-dimensional Euclidean space. It is an application of our partitioning theorem together with the refined bounds for the number of connected components of a semi-algebraic set by Barone and Basu.

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Acknowledgments

We thank Zuzana Safernová/Patáková, Micha Sharir, Noam Solomon and Joshua Zahl for useful discussions and pointers to the literature. We also thank the anonymous referees for their remarks and corrections, which have significantly improved this paper. Part of this work was done while the authors met at the Institute for Pure and Applied Mathematics (IPAM) during the Spring 2014 research program “Algebraic Techniques for Combinatorial and Computational Geometry”. Basu and Sombra were partially supported by the IPAM research program “Algebraic Techniques for Combinatorial and Computational Geometry”. Basu was also partially supported by NSF grants CCF-0915954, CCF-1319080 and DMS-1161629. Sombra was also partially supported by the MINECO research project MTM2012-38122-C03-02.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47906, USA

    Saugata Basu

  2. ICREA & Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Martín Sombra

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  1. Saugata Basu
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  2. Martín Sombra
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Correspondence to Martín Sombra.

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Basu, S., Sombra, M. Polynomial Partitioning on Varieties of Codimension Two and Point-Hypersurface Incidences in Four Dimensions. Discrete Comput Geom 55, 158–184 (2016). https://doi.org/10.1007/s00454-015-9736-4

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  • DOI: https://doi.org/10.1007/s00454-015-9736-4

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