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Multilevel Polynomial Partitions and Simplified Range Searching

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Abstract

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.

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Notes

  1. More precisely, these are affine algebraic varieties, while other kinds of algebraic varieties, such as projective or quasiprojective ones, are often considered in the literature. Here, with a single exception, it suffices for us to consider the affine case.

  2. The Krull dimension of a ring R is the largest n such that there exists a chain \(I_0\subsetneq I_1\subsetneq \cdots \subsetneq I_n\) of nested prime ideals in R.

  3. A polynomial h is reducible w.r.t. I if \({\mathsf {supp}}(h) \cap \langle {\ell m}(I)\rangle \ne \emptyset ,\) where the support of h is a set of all monomials occurring in h (i.e., having nonzero coefficient), and \(\langle {\ell m}(I)\rangle = \langle {\ell m}(f) :f \in I\rangle \) is an ideal of all leading monomials of I, where leading monomial \({\ell m}(f)\) is the largest monomial occurring in f. We note that a monic monomial is reducible if and only if it is different from its normal form.

  4. \(h+I= \{h+f:f \in I\}.\)

  5. We note that the algorithm by [27] requires the monomial ordering given by rational weight matrix. The weight matrix of lexicographic ordering consists just of zeros and ones, and hence it is rational. See [27] for details.

  6. A monomial h is minimally reducible w.r.t. I if it is reducible w.r.t. I but none of direct divisors is reducible w.r.t. I, where direct divisors of a term are just those terms where the exponent vector is smaller by 1 in exactly one coordinate and equal in all others.

  7. A ring S is an integral extension of a subring \(R\subseteq S\) if all elements of S are roots of monic polynomials in R[x].

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Acknowledgments

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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Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Prague 1, Czech Republic

    Jiří Matoušek

  2. Department of Computer Science, ETH Zurich, 8092, Zurich, Switzerland

    Jiří Matoušek

  3. Department of Applied Mathematics and Computer Science Institute, Charles University, Malostranské nám. 25, 118 00, Prague 1, Czech Republic

    Zuzana Patáková

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Matoušek, J., Patáková, Z. Multilevel Polynomial Partitions and Simplified Range Searching. Discrete Comput Geom 54, 22–41 (2015). https://doi.org/10.1007/s00454-015-9701-2

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