Discrete & Computational Geometry

, Volume 54, Issue 1, pp 22–41 | Cite as

Multilevel Polynomial Partitions and Simplified Range Searching

  • Jiří Matoušek
  • Zuzana PatákováEmail author


The polynomial partitioning method of Guth and Katz (arXiv:1011.4105) has numerous applications in discrete and computational geometry. It partitions a given n-point set \(P\subset {\mathbb {R}}^d\) using the zero set Z(f) of a suitable d-variate polynomial f. Applications of this result are often complicated by the problem, “What should be done with the points of P lying within Z(f)?” A natural approach is to partition these points with another polynomial and continue further in a similar manner. So far this has been pursued with limited success—several authors managed to construct and apply a second partitioning polynomial, but further progress has been prevented by technical obstacles. We provide a polynomial partitioning method with up to d polynomials in dimension d, which allows for a complete decomposition of the given point set. We apply it to obtain a new algorithm for the semialgebraic range searching problem. Our algorithm has running time bounds similar to a recent algorithm by Agarwal et al. (SIAM J Comput 42:2039–2062, 2013), but it is simpler both conceptually and technically. While this paper has been in preparation, Basu and Sombra, as well as Fox, Pach, Sheffer, Suk, and Zahl, obtained results concerning polynomial partitions which overlap with ours to some extent.


Irreducible Component Algebraic Variety Computational Geometry Real Polynomial Arbitrary Real Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The research was supported by the ERC Advanced Grant No. 267165. Z. Patáková was partially supported by the Project CE-ITI (GACR P202/12/G061) of the Czech Science Foundation and by the Charles University Grants SVV-2014-260103 and GAUK 690214. We would like to thank Josh Zahl for pointing out mistakes in an earlier version of this paper, Saugata Basu for providing a draft of his recent work with Sombra and useful advice, Erich Kaltofen for kindly answering our questions concerning polynomial factorization, and Pavel Paták, Edgardo Roldán Pensado, Martín Sombra, and Martin Tancer for enlightening discussions.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Applied MathematicsCharles UniversityPrague 1Czech Republic
  2. 2.Department of Computer ScienceETH ZurichZurichSwitzerland
  3. 3.Department of Applied Mathematics and Computer Science InstituteCharles UniversityPrague 1Czech Republic

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