Discrete & Computational Geometry

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

Multilevel Polynomial Partitions and Simplified Range Searching



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.


  1. 1.
    Agarwal, P.K., Erickson, J.: Geometric range searching and its relatives. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Discrete and Computational Geometry: Ten Years Later, pp. 1–56. American Mathematical Society, Providence, RI (1998)Google Scholar
  2. 2.
    Agarwal, P.K., Matoušek, J., Sharir, M.: On range searching with semialgebraic sets II. SIAM J. Comput. 42(6), 2039–2062 (2013)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Barone, S., Basu, S.: Refined bounds on the number of connected components of sign conditions on a variety. Discrete Comput. Geom. 47(3), 577–597 (2012)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Barone, S., Basu, S.: On a real analogue of Bezout inequality and the number of connected components of sign conditions. Preprint, arXiv:1303.1577v2 (2013)
  5. 5.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2003)Google Scholar
  6. 6.
    Basu, S., Sombra, M.: Polynomial partitioning on varieties and point-hypersurface incidences in four dimensions. Preprint, arXiv:1406.2144 (2014)
  7. 7.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin etc. Translated from the French, revised and updated edition (1998)Google Scholar
  8. 8.
    Chan, T.M.: Optimal partition trees. Discrete Comput. Geom. 47(4), 661–690 (2012)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chazelle, B.: Cuttings. In: Mehta, D.P., Sahni, S. (eds.) Handbook of Data Structures and Applications. Chapman & Hall/CRC, Boca Raton (2005)Google Scholar
  10. 10.
    Clarkson, K.L.: New applications of random sampling in computational geometry. Discrete Comput. Geom. 2, 195–222 (1987)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms. Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2007)Google Scholar
  12. 12.
    Decker, W., Pfister, G.: A first course in computational algebraic geometry. African Institute of Mathematics (AIMS) Library Series. Cambridge University Press, Cambridge (2013)CrossRefMATHGoogle Scholar
  13. 13.
    Dubé, T.W.: The structure of polynomial ideals and Gröbner bases. SIAM J. Comput. 19(4), 750–775 (1990)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Elkadi, M., Mourrain, B.: A new algorithm for the geometric decomposition of a variety. In: Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation, pp. 9–16, ACM, New York (1999)Google Scholar
  15. 15.
    Fox, J., Pach, J., Sheffer, A., Suk, A., Zahl, J.: A semi-algebraic version of Zarankiewicz’s problem. Preprint, arXiv:1407.5705 (2014)
  16. 16.
    Guth, L.: Distinct distance estimates and low degree polynomial partitioning. Discrete Comput. Geom. 53(2), 428–444 (2015)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Guth, L., Katz, N.H.: On the Erdős distinct distances problem in the plane. Ann. Math. 181, 155–190 (2015)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Harris, J.: Algebraic Geometry (A First Course). Springer, Berlin (1992)MATHGoogle Scholar
  19. 19.
    Hartshorne, R.: Algebraic Geometry. Springer, New York (1977)CrossRefMATHGoogle Scholar
  20. 20.
    Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Heintz, J.: Definability and fast quantifier elimination in algebraically closed fields. Theoret. Comput. Sci. 24(3), 239–277 (1983). Corrigendum ibid. 39 1983: 2–3CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hermann, G.: Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Math. Ann. 95(1), 736–788 (1926)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)MATHGoogle Scholar
  24. 24.
    Kaltofen, E.: Polynomial factorization 1987–1991. LATIN ’92 (São Paulo, 1992). Lecture Notes in Computer Science, pp. 294–313. Springer, Berlin (1992)Google Scholar
  25. 25.
    Kaplan, H., Matoušek, J., Sharir, M.: Simple proofs of classical theorems in discrete geometry via the Guth-Katz polynomial partitioning technique. Discrete Comput. Geom. 48(3), 499–517 (2012)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Kaplan, H., Matoušek, J., Safernová, Z., Sharir, M.: Unit distances in three dimensions. Comb. Probab. Comput. 21(4), 597–610 (2012)CrossRefMATHGoogle Scholar
  27. 27.
    Kühnle, K., Mayr, E.W.: Exponential space computation of Gröbner bases. In: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation. ISSAC ’96, Zürich, Switzerland, 24–26 July, 1996, pp. 62–71. ACM, New York (1996)Google Scholar
  28. 28.
    Matoušek, J.: Efficient partition trees. Discrete Comput. Geom. 8, 315–334 (1992)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Matoušek, J.: Geometric range searching. ACM Comput. Surveys 26, 421–461 (1995)Google Scholar
  30. 30.
    Mayr, E.W., Meyer, A.R.: The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. Math. 46(3), 305–329 (1982)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Mayr, E.W. Ritscher, S.: Space-efficient Gröbner basis computation without degree bounds. In: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, pp. 257–264. ACM, New York (2011)Google Scholar
  32. 32.
    Sharir, M., Solomon, N.: Incidences between points and lines in four dimensions. In: Proceedings of 30th ACM Symposium on Computational Geometry (2014)Google Scholar
  33. 33.
    Sharir, M., Sheffer, A., Zahl, J.:. Improved bounds for incidences between points and circles. Preprint arXiv:1208.0053 (2012)
  34. 34.
    Solymosi, J., Tao, T.: An incidence theorem in higher dimensions. Discrete Comput. Geom. 48(2), 255–280 (2012)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Whitney, H.: Elementary structure of real algebraic varieties. Ann. Math. 2(66), 545–556 (1957)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Wang, H., Yang, B., Zhang, R.:. Bounds of incidences between points and algebraic curves. Preprint, arXiv:1308.0861 (2013)
  37. 37.
    Zahl, J.: A Szemeredi-Trotter type theorem in \({\mathbb{R}}^4\). Preprint, arXiv:1203.4600 (2012)
  38. 38.
    Zahl, J.: An improved bound on the number of point-surface incidences in three dimensions. Contrib. Discrete Math. 8(1), 100–121 (2013)MathSciNetGoogle Scholar

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

Personalised recommendations