The Szemerédi-Trotter theorem in the complex plane

Abstract

It is shown that n points and e lines in the complex Euclidean plane ℂ2 determine O(n 2/3 e 2/3 + n + e) point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemerédi and Trotter about point-line incidences in the real Euclidean plane ℝ2.

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References

  1. [1]

    J. Beck: On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry, Combinatorica 3 (1983), 281–197.

    Article  MATH  MathSciNet  Google Scholar 

  2. [2]

    K. L. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl: Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), 99–160.

    Article  MATH  MathSciNet  Google Scholar 

  3. [3]

    H. Edelsbrunner: Algorithms in Combinatorial Geometry, EATCS Monographs in Theoretical Computer Science, Springer, 1987.

    Google Scholar 

  4. [4]

    H. Edelsbrunner, L. Guibas and M. Sharir: The complexity of many cells in three dimensional arrangements, Discrete Comput. Geom. 5 (1990), 197–216.

    Article  MATH  MathSciNet  Google Scholar 

  5. [5]

    Gy. Elekes: A combinatorial problem on polynomials, Discrete Comput. Geom. 19 (1998), 383–389.

    Article  MATH  MathSciNet  Google Scholar 

  6. [6]

    Gy. Elekes: On linear combinatorics I, Concurrency an algebraic approach, Combinatorica 17 (1997), 447–458.

    Article  MATH  MathSciNet  Google Scholar 

  7. [7]

    P. Erdős: Problems and results in combinatorial geometry, in: Discrete geometry and convexity (New York, 1982), vol. 440 of Ann. New York Acad. Sci., 1985, 1–11.

    Google Scholar 

  8. [8]

    G. H. Golub and C. F. Van Loan: Matrix computations, The Johns Hopkins Univ. Press (2nd ed.), Baltimore, MD, 1989, 584–586.

    Google Scholar 

  9. [9]

    P. Henrici: Applied and computational complex analysis, Vol. 1, John Wiley & Sons, New York, NY, 1974, 307–314.

    MathSciNet  Google Scholar 

  10. [10]

    K. Leichtweiss: Zur Riemannischen Geometrie in Grassmannschen Mannigfaltigkeiten, Math. Z. 76 (1961), 334–336.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    R. Narasimhan: Analysis on real and complex manifolds, Elsevier (3rd ed.), Amsterdam, 1985, 66–69.

    Google Scholar 

  12. [12]

    J. Pach and P. K. Agarwal: Combinatorial Geometry, Wiley, New York, 1995, 180–181.

    Google Scholar 

  13. [13]

    J. Pach, R. RadoičiĆ, G. Tardos and G. Tóth: Improving the Crossing Lemma by finding more crossings in sparse graphs, Discrete Comput. Geom. 36 (4)(2006), 527–552.

    Article  MATH  MathSciNet  Google Scholar 

  14. [14]

    J. Pach and M. Sharir: Repeated angles in the plane and related problems, J. Comb. Theory, Ser. A 59 (1992), 12–22.

    Article  MATH  MathSciNet  Google Scholar 

  15. [15]

    J. Pach and M. Sharir: On the number of incidences between points and curves, Combinatorics, Probability and Computing 7 (1998), 121–127.

    Article  MATH  MathSciNet  Google Scholar 

  16. [16]

    J. Pach and M. Sharir: Geometric incidences, in: Towards a theory of geometric graphs, vol. 342 of Contemporary Mathematics, AMS, Providence, RI, 2004, 185–224.

    MathSciNet  Google Scholar 

  17. [17]

    J. Pach and G. Tóth: Graphs drawn with few crossings per edge, Combinatorica 17 (1997), 427–439.

    Article  MATH  MathSciNet  Google Scholar 

  18. [18]

    J. Solymosi: On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), 491–494.

    Article  MATH  MathSciNet  Google Scholar 

  19. [19]

    L. Szekely: Crossing numbers and hard Erdős problems in discrete geometry, RCombinatorics, Probability and Computing 6 (1997), 353–358.

    Article  MATH  MathSciNet  Google Scholar 

  20. [20]

    E. Szemerédi and W. T. Trotter: Extremal problems in discrete geometry, Combinatorica 3 (1983), 381–392.

    Article  MATH  MathSciNet  Google Scholar 

  21. [21]

    E. Szemerédi and W. T. Trotter: A combinatorial distinction between the Euclidean and projective planes, European Journal of Combinatorics 4 (1983), 385–394.

    Article  MATH  MathSciNet  Google Scholar 

  22. [22]

    Y-C. Wong: Differential geometry of Grassmann manifolds, Proc. Nat. Acad. Sci. U.S.A. 47 (1967), 189–594.

    Google Scholar 

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Correspondence to Csaba D. Tóth.

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Research on this paper was conducted at the Eötvös Loránd University, Budapest.

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Tóth, C.D. The Szemerédi-Trotter theorem in the complex plane. Combinatorica 35, 95–126 (2015). https://doi.org/10.1007/s00493-014-2686-2

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Mathematics Subject Classication (2000)

  • 05D99
  • 52C35