Duke Mathematical Journal

Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited

Orit E. Raz, Micha Sharir, and Frank De Zeeuw

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Abstract

Let FC[x,y,z] be a constant-degree polynomial, and let A,B,CC be finite sets of size n. We show that F vanishes on at most O(n11/6) points of the Cartesian product A×B×C, unless F has a special group-related form. This improves a theorem of Elekes and Szabó and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over R, and a similar statement holds when A,B,C have different sizes (with a more involved bound replacing O(n11/6)). This result provides a unified tool for improving bounds in various Erdős-type problems in combinatorial geometry, and we discuss several applications of this kind.

Article information

Source
Duke Math. J., Volume 165, Number 18 (2016), 3517-3566.

Dates
Received: 28 April 2015
Revised: 18 February 2016
First available in Project Euclid: 7 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1473275860

Digital Object Identifier
doi:10.1215/00127094-3674103

Mathematical Reviews number (MathSciNet)
MR3577370

Zentralblatt MATH identifier
1365.52023

Subjects
Primary: 52C10: Erdős problems and related topics of discrete geometry [See also 11Hxx]
Secondary: 05D99: None of the above, but in this section

Keywords
combinatorial geometry incidences polynomials

Citation

Raz, Orit E.; Sharir, Micha; De Zeeuw, Frank. Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited. Duke Math. J. 165 (2016), no. 18, 3517--3566. doi:10.1215/00127094-3674103. https://projecteuclid.org/euclid.dmj/1473275860


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References

  • [1] S. Basu, R. Pollack, and M.-F. Roy, Algorithms in Real Algebraic Geometry, Algorithms Comput. Math. 10, Springer, Berlin, 2003.
  • [2] M. Charalambides, Distinct distances on curves via rigidity, Discrete Comput. Geom. 51 (2014), 666–701.
  • [3] G. Elekes, A combinatorial problem on polynomials, Discrete Comput. Geom. 19 (1998), 383–389.
  • [4] G. Elekes, A note on the number of distinct distances, Period. Math. Hungar. 38 (1999), 173–177.
  • [5] G. Elekes, “SUMS versus PRODUCTS in number theory, algebra and Erdős geometry” in Paul Erdős and His Mathematics, II, Bolyai Soc. Math. Stud. 11, Springer, Berlin, 2002, 241–290.
  • [6] G. Elekes and L. Rónyai, A combinatorial problem on polynomials and rational functions, J. Combinat. Theory Ser. A 89 (2000), 1–20.
  • [7] G. Elekes, M. Simonovits, and E. Szabó, A combinatorial distinction between unit circles and straight lines: How many coincidences can they have?, Combinat. Probab. Comput. 18 (2009), 691–705.
  • [8] G. Elekes and E. Szabó, How to find groups? (And how to use them in Erdős geometry?), Combinatorica 32 (2012), 537–571.
  • [9] G. Elekes and E. Szabó, On triple lines and cubic curves: The orchard problem revisited, preprint, arXiv:1302.5777v1 [math.CO].
  • [10] K. Fritzsche and H. Grauert, From Holomorphic Functions to Complex Manifolds, Grad. Texts in Math. 213, Springer, New York, 2002.
  • [11] J. Harris, Algebraic Geometry: A First Course, Grad. Texts in Math. 133, Springer, New York, 1992.
  • [12] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • [13] J. Heintz, Definability and fast quantifier elimination in algebraically closed fields, Theoret. Comput. Sci. 24 (1983), 239–277.
  • [14] S. Lang, Complex Analysis, Grad. Texts in Math. 103, Springer, New York, 1999.
  • [15] J. Pach and F. De Zeeuw, Distinct distances on algebraic curves in the plane, preprint, arXiv:1308.0177 [math.MG].
  • [16] O. E. Raz, M. Sharir, and J. Solymosi, On triple intersections of three families of unit circles, Discrete Comput. Geom. 54 (2015), 930–953.
  • [17] O. E. Raz, M. Sharir, and J. Solymosi, Polynomials vanishing on grids: The Elekes-Rónyai problem revisited, preprint, arXiv:1401.7419 [cs.CG].
  • [18] J. Schwartz, Fast probabilistic algorithms for verification of polynomial identities, J. ACM 27 (1980), 701–717.
  • [19] M. Sharir, A. Sheffer, and J. Solymosi, Distinct distances on two lines, J. Combin. Theory Ser. A 120 (2013), 1732–1736.
  • [20] M. Sharir and J. Solymosi, Distinct distances from three points, Combin. Probab. Comput. 25 (2016), 623–632.
  • [21] A. Sheffer, E. Szabó, and J. Zahl, Point-curve incidences in the complex plane, preprint, arXiv:1502.07003v3 [math.CO].
  • [22] J. Solymosi and F. De Zeeuw, Incidence bounds for complex algebraic curves on Cartesian products, preprint, arXiv:1502.05304v2 [math.CO].
  • [23] J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput. Geom. 48 (2012), 255–280.
  • [24] E. Szemerédi and W. T. Trotter, Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), 381–392.
  • [25] T. Tao, Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets, Contrib. Discrete Math. 10 (2015), 22–98.
  • [26] T. Tao, Bézout’s inequality, preprint, http://terrytao.wordpress.com/2011/03/23/bezouts-inequality (accessed 22July 2016).
  • [27] C. D. Tóth, The Szemerédi-Trotter theorem in the complex plane, Combinatorica 35 (2015), 95–126.
  • [28] H. Wang, Exposition of Elekes Szabó paper, preprint, arXiv:1512.04998v1 [math.CO].
  • [29] J. Zahl, A Szemerédi-Trotter type theorem in $\mathbb{R}^{4}$, Discrete Comput. Geom. 54 (2015), 513–572.
  • [30] R. Zippel, An explicit separation of relativised random polynomial time and relativised deterministic polynomial time, Inform. Process. Lett. 33 (1989), 207–212.