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Incidence Bounds for Complex Algebraic Curves on Cartesian Products

  • József SolymosiEmail author
  • Frank de Zeeuw
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)

Abstract

We prove bounds on the number of incidences between a set of algebraic curves in \({\mathbb C}^2\) and a Cartesian product \(A\times B\) with finite sets \(A,B\subset {\mathbb C}\). Similar bounds are known under various restrictive conditions, but we show that the Cartesian product assumption leads to a simpler proof and lets us remove these conditions. This assumption holds in a number of interesting applications, and with our bound these applications can be extended from \({\mathbb R}\) to \({\mathbb C}\). We also obtain more precise information in the bound, which is used in several recent papers (Raz et al., 31st international symposium on computational geometry (SoCG 2015), pp 522–536, 2015, [17], Valculescu, de Zeeuw, Distinct values of bilinear forms on algebraic curves, 2014, [25]). Our proof works via an incidence bound for surfaces in \({\mathbb R}^4\), which has its own applications (Raz, Sharir, 31st international symposium on computational geometry (SoCG 2015), pp 569–583, 2015, [15]). The proof is a new application of the polynomial partitioning technique introduced by Guth and Katz (Ann Math 181: 155–190, 2015, [11]).

References

  1. 1.
    S. Barone, S. Basu, On a real analogue of Bezout inequality and the number of connected components of sign conditions, in Proceedings of the London Mathematical Society, vol. 112, n. 1 (1 January 2016), pp. 115–145, arXiv:1303.1577
  2. 2.
    S. Basu, R. Pollack, M.-F. Roy, Algorithms in Real Algebraic Geometry (Springer, Berlin, 2003)zbMATHGoogle Scholar
  3. 3.
    S. Basu, M. Sombra, Polynomial partitioning on varieties and point-hypersurface incidences in four dimensions, Discrete Comput. Geom. 55(1), 158–184 (January 2016), arXiv:1406.2144
  4. 4.
    J. Bochnak, M. Coste, M.-F. Roy, Real Algebraic Geometry (Springer, Berlin, 1998)CrossRefGoogle Scholar
  5. 5.
    B. Bollobás, Modern Graph Theory (Springer, Berlin, 1998)CrossRefGoogle Scholar
  6. 6.
    G. Elekes, On the Dimension of Finite Point Sets II. “Das Budapester Programm" (2011), arXiv:1109.0636
  7. 7.
    G. Elekes, SUMS versus PRODUCTS in Number Theory. Algebra and Erdős Geometry, Paul Erdős and his Mathematics II, Bolyai Society Mathematical Studies 11, 241–290 (2002)zbMATHGoogle Scholar
  8. 8.
    G. Elekes, L. Rónyai, A combinatorial problem on polynomials and rational functions. J. Comb. Theory, Ser. A 89, 1–20 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    G. Elekes, M. Nathanson, I.Z. Ruzsa, Convexity and sumsets. J. Number Theory 83, 194–201 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    G. Fischer, Plane Algebraic Curves (American Mathematical Society, Providence, 2001)zbMATHGoogle Scholar
  11. 11.
    L. Guth, N.H. Katz, On the Erdős distinct distances problem in the plane. Ann. Math. 181, 155–190 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    J. Harris, Algebraic Geometry: A First Course (Springer, Berlin, 1992)CrossRefGoogle Scholar
  13. 13.
    J. Heintz, Definability and fast quantifier elimination in algebraically closed fields. Theor. Comput. Sci. 24, 239–277 (1983)MathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Pach, M. Sharir, On the number of incidences between points and curves. Comb. Probab. Comput. 7, 121–127 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    O.E. Raz, M. Sharir, The number of unit-area triangles in the plane: Theme and variations. Combinatorica 37(6), 1221–1240 (December 2017). Also in arXiv:1501.00379MathSciNetCrossRefGoogle Scholar
  16. 16.
    O.E. Raz, M. Sharir, J. Solymosi, Polynomials vanishing on grids: The Elekes-Rónyai problem revisited, in Proceedings of the Thirtieth Annual Symposium on Computational Geometry (2014), pp. 251–260, arXiv:1401.7419
  17. 17.
    O.E. Raz, M. Sharir, F. de Zeeuw, Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited, in 31st International Symposium on Computational Geometry (SoCG 2015) (2015), pp. 522–536. Also in arXiv:1504.05012
  18. 18.
    M. Sharir, A. Sheffer, J. Solymosi, Distinct distances on two lines. J. Comb. Theory, Ser. A 120, 1732–1736 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Sheffer, E. Szabó, J. Zahl, Point-curve incidences in the complex plane. Combinatorica 38(2), 487–499 (April 2018), arXiv:1502.07003MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Solymosi, On the number of sums and products. Bull. Lond. Math. Soc. 37, 491–494 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    J. Solymosi, T. Tao, An incidence theorem in higher dimensions. Discrete Comput. Geom. 48, 255–280 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    J. Solymosi, G. Tardos, On the number of k-rich transformations, in Proceedings of the Twenty-Third Annual Symposium on Computational Geometry (2007), pp. 227–231Google Scholar
  23. 23.
    J. Solymosi, V. Vu, Distinct distances in high dimensional homogeneous sets, Towards a theory of geometric graphs. Contemp. Math. 342, 259–268 (American Mathematical Society, 2004)Google Scholar
  24. 24.
    C.D. Tóth, The Szemerédi-Trotter theorem in the complex plane. Combinatorica 35, 95–126 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    C. Valculescu, F. de Zeeuw, Distinct values of bilinear forms on algebraic curves. Contributions to Discrete Mathematics, 11(1), (July 2016). Also in arXiv:1403.3867
  26. 26.
    J. Zahl, A Szemerédi-Trotter type theorem in \({\mathbb{R}}^4\). Discrete Comput. Geom. 54, 513–572 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.VancouverCanada
  2. 2.EPFLLausanneSwitzerland

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