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A Survey of Elekes-Rónyai-Type Problems

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New Trends in Intuitive Geometry

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 27))

Abstract

We give an overview of recent progress around a problem introduced by Elekes and Rónyai. The prototype problem is to show that a polynomial \(f\in \mathbb R[x,y]\) has a large image on a Cartesian product \(A\times B\subset \mathbb R^2\), unless f has a group-related special form. We discuss this problem and a number of variants and generalizations. This includes the Elekes-Szabó problem, which generalizes the Elekes-Rónyai problem to a question about an upper bound on the intersection of an algebraic surface with a Cartesian product, and curve variants, where we ask the same questions for Cartesian products of finite subsets of algebraic curves. These problems lie at the crossroads of combinatorics, algebra, and geometry: They ask combinatorial questions about algebraic objects, whose answers turn out to have applications to geometric questions involving basic objects like distances, lines, and circles, as well as to sum-product-type questions from additive combinatorics. As part of a recent surge of algebraic techniques in combinatorial geometry, a number of quantitative and qualitative steps have been made within this framework. Nevertheless, many tantalizing open questions remain.

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Notes

  1. 1.

    The result in [28, Proposition 8.3] has the same proof setup as [47], but it appears to be somewhat isolated; it makes no reference to [17], and in turn is not referred to in [43, 47]. This may be because [28] primarily concerns expansion bounds over finite fields. The arXiv publication date of [47] is half a year before that of [28].

  2. 2.

    The maximum of the degrees of the numerator and denominator, assuming that these do not have a common factor.

  3. 3.

    The chronology is somewhat confusing here. The paper [17] was published in 2000, and [15] in 1999. However, [15] refers back to [17] and makes it clear that [15] is an improvement on a special case of [17].

  4. 4.

    A preprint version of [26] became available in 2010.

  5. 5.

    The proof in [43] did not directly use the bound from [36], but rather adapted the proof from [36] to the incidence situation in [43].

  6. 6.

    The word “grid” is often used in this context, but may lead to confusion with integer grids.

  7. 7.

    We write Z(F) for the zero set of a polynomial F, i.e., the set of points at which F vanishes.

  8. 8.

    The publication year of [19] is 2012, but an essentially complete version of the paper existed much earlier; Elekes [10] referred to the result in 2002.

  9. 9.

    Earlier versions of [19] claimed a bound of the form \(O(n^{2-\eta })\) for an absolute \(\eta >0\), and this was restated in [10]. But the published version states the theorem with \(\eta _d\) depending on d.

  10. 10.

    The new algebraic methods introduced in [26] indirectly led to the improvement of [47] in Theorem 1.4, and thus to many of the recent results in this survey.

  11. 11.

    This is a special case of a more general object from category theory; what we call a fiber product here is sometimes called a set-theoretic fiber product, or also a relative product.

  12. 12.

    As in Sect. 1.3, \(D(p,q) = (p_x-q_x)^2+(p_y-q_y)^2\) is the squared Euclidean distance function.

  13. 13.

    We say that two curves are linearly equivalent if there is a linear transformation \((x,y)\mapsto (ax+by,cx+dy)\) that gives a bijection between the point sets of the curves.

  14. 14.

    This brings into question if “Schwartz–Zippel” is the right name, but it has become standard in combinatorics.

References

  1. E. Aksoy Yazici, B. Murphy, M. Rudnev, I. Shkredov, Growth estimates in positive characteristic via collisions. International Mathematics Research Notices, 2017(23), 7148–7189 (1 December 2017), arXiv:1512.06613

  2. J. Beck, On the lattice property of the plane and some problems of Dirac, Motzkin, and Erdős in combinatorial geometry. Combinatorica 3, 281–297 (1983)

    Article  MathSciNet  Google Scholar 

  3. J. Bourgain, More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1, 1–32 (2005)

    Article  MathSciNet  Google Scholar 

  4. J. Bourgain, N. Katz, T. Tao, A sum-product estimate in finite fields, and applications. Geom. Funct. Anal. 14, 27–57 (2004)

    Article  MathSciNet  Google Scholar 

  5. P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry (Springer, Berlin, 2005)

    Google Scholar 

  6. A. Bronner, A. Sheffer, M. Sharir, Distinct distances on a line and a curve (2016) [manuscript]

    Google Scholar 

  7. B. Bukh, J. Tsimerman, Sum-product estimates for rational functions. Proc. Lond. Math. Soc. 104, 1–26 (2012)

    Article  MathSciNet  Google Scholar 

  8. M. Charalambides, Distinct distances on curves via rigidity. Discret. Comput. Geom. 51, 666–701 (2014)

    Article  MathSciNet  Google Scholar 

  9. F. de Zeeuw, A course in algebraic combinatorial geometry, lecture notes available from the author’s website http://dcg.epfl.ch/page-84876-en.html (2015)

  10. G. Elekes, in SUMS versus PRODUCTS in Number Theory Algebra and Erdős Geometry, Paul Erdős and his Mathematics II, Bolyai Society Mathematical Studies, vol. 11 (2002), pp. 241–290

    Google Scholar 

  11. G. Elekes, E. Szabó, On triple lines and cubic curves: the orchard problem revisited, arXiv:1302.5777 (2013)

  12. G. Elekes, Circle grids and bipartite graphs of distances. Combinatorica 15, 167–174 (1995)

    Article  MathSciNet  Google Scholar 

  13. G. Elekes, On the number of sums and products. Acta Arith. 81, 365–367 (1997)

    Article  MathSciNet  Google Scholar 

  14. G. Elekes, A combinatorial problem on polynomials. Discret. Comput. Geom. 19, 383–389 (1998)

    Article  MathSciNet  Google Scholar 

  15. G. Elekes, A note on the number of distinct distances. Period. Math. Hung. 38, 173–177 (1999)

    Article  MathSciNet  Google Scholar 

  16. G. Elekes, On linear combinatorics III, few directions and distorted lattices. Combinatorica 1, 43–53 (1999)

    Article  MathSciNet  Google Scholar 

  17. G. Elekes, L. Rónyai, A combinatorial problem on polynomials and rational functions. J. Comb. Theory Ser. A 89, 1–20 (2000)

    Article  MathSciNet  Google Scholar 

  18. G. Elekes, M. Sharir, Incidences in three dimensions and distinct distances in the plane. Comb. Probab. Comput. 20, 571–608 (2011)

    Article  MathSciNet  Google Scholar 

  19. G. Elekes, E. Szabó, How to find groups? (And how to use them in Erdős geometry?). Combinatorica 32, 537–571 (2012)

    Article  MathSciNet  Google Scholar 

  20. G. Elekes, M.B. Nathanson, I.Z. Ruzsa, Convexity and sumsets. J. Number Theory 83, 194–201 (1999)

    Article  MathSciNet  Google Scholar 

  21. G. Elekes, M. Simonovits, E. Szabó, A combinatorial distinction between unit circles and straight lines: how many coincidences can they have? Comb. Probab. Comput. 18, 691–705 (2009)

    Article  MathSciNet  Google Scholar 

  22. P. Erdős, E. Szemerédi, On sums and products of integers, in Studies in Pure Mathematics (Birkhäuser, 1983), pp. 213–218

    Google Scholar 

  23. P. Erdős, On sets of distances of \(n\) points. Am. Math. Mon. 53, 248–250 (1946)

    Article  MathSciNet  Google Scholar 

  24. P. Erdős, G. Purdy, Some extremal problems in geometry. J. Comb. Theory 10, 246–252 (1971)

    Article  MathSciNet  Google Scholar 

  25. B. Green, T. Tao, On sets defining few ordinary lines. Discret. Comput. Geom. 50, 409–468 (2013)

    Article  MathSciNet  Google Scholar 

  26. L. Guth, N.H. Katz, On the Erdős distinct distances problem in the plane. Ann. Math. 181, 155–190 (2015)

    Article  MathSciNet  Google Scholar 

  27. R. Hartshorne, Algebraic Geometry (Springer, Berlin, 1977)

    Book  Google Scholar 

  28. N. Hegyvári, F. Hennecart, Conditional expanding bounds for two-variable functions over prime fields. Eur. J. Comb. 34, 1365–1382 (2013)

    Article  MathSciNet  Google Scholar 

  29. S. Konyagin, I. Shkredov, On sum sets of sets, having small product set, in Proceedings of the Steklov Institute of Mathematics, vol. 290, n.1 (August 2015), pp. 288–299, arXiv:1503.05771

    Article  MathSciNet  Google Scholar 

  30. S. Lang, A. Weil, Number of points of varieties in finite fields. Am. J. Math. 76, 819–827 (1954)

    Article  MathSciNet  Google Scholar 

  31. R.J. Lipton, blog post (2009), http://rjlipton.wordpress.com/2009/11/30//the-curious-history-of-the-schwartz-zippel-lemma/

  32. B. Lund, A. Sheffer, F. de Zeeuw, Bisector energy and few distinct distances, in 31st International Symposium on Computational Geometry (SoCG 2015) (2015), pp. 537–552, arXiv:1411.6868

  33. J. Matoušek, Lectures on Discrete Geometry (Springer, Berlin, 2002)

    Google Scholar 

  34. H. Nassajian Mojarrad, T. Pham, C. Valculescu, F. de Zeeuw, Schwartz–Zippel Bounds for Two-dimensional Products (2015), arXiv:1507.08181

  35. J. Pach, F. de Zeeuw, Distinct distances on algebraic curves in the plane, Comb. Probab. Comput. 26(1), 99–117 (2017), arXiv:1308.0177; Proceedings of the Thirtieth Annual Symposium on Computational geometry (2014), pp. 549–557

  36. J. Pach, M. Sharir, On the number of incidences between points and curves. Comb. Probab. Comput. 7, 121–127 (1998)

    Article  MathSciNet  Google Scholar 

  37. O.E. Raz, A Note on Distinct Distance Subsets (2016), arXiv:1603.00740

  38. O.E. Raz, M. Sharir, Rigidity of complete bipartite graphs in the plane (2016) [manuscript]

    Google Scholar 

  39. O.E. Raz, M. Sharir, The number of unit-area triangles in the plane: Theme and variations. Combinatorica 37(6), 1221–1240 (December 2017), arXiv:1501.00379

    Article  MathSciNet  Google Scholar 

  40. O.E. Raz, M. Sharir, F. de Zeeuw, Polynomials vanishing on cartesian products: The Elekes–Szabó theorem revisited, Duke Math. J. 165(18), 3517–3566 (2016), arXiv:1504.05012; in 31st International Symposium on Computational Geometry (SoCG 2015) (2015), pp. 522–536

    Article  MathSciNet  Google Scholar 

  41. O.E. Raz, M. Sharir, F. de Zeeuw, The Elekes–Szabó theorem in four dimensions. Israel Journal of Mathematics 227(2), 663–690 (August 2018) [manuscript]

    Article  MathSciNet  Google Scholar 

  42. O.E. Raz, M. Sharir, J. Solymosi, On triple intersections of three families of unit circles, Discret. Comput. Geom. 54, 930–953 (2015); in Proceedings of the Thirtieth Annual Symposium on Computational Geometry (2014), pp. 198–205

    Article  MathSciNet  Google Scholar 

  43. O.E. Raz, M. Sharir, J. Solymosi, Polynomials vanishing on grids: the Elekes–Rónyai problem revisited. Am. J. Math. arXiv:1401.7419; in Proceedings of the thirtieth annual symposium on Computational geometry (2014), pp. 251–260

  44. R. Schwartz, J. Solymosi, F. de Zeeuw, Extensions of a result of Elekes and Rónyai. J. Comb. Theory Ser. A 120, 1695–1713 (2013)

    Article  Google Scholar 

  45. M. Sharir, S. Smorodinsky, C. Valculescu, F. de Zeeuw, Distinct distances between points and lines. Comput. Geom. 69, 2–15 (2018), arXiv:1512.09006

    Article  MathSciNet  Google Scholar 

  46. M. Sharir, J. Solymosi, Distinct distances from three points, Comb. Probab. Comput. 25(4), 623–632 (2016), arXiv:1308.0814

    Article  MathSciNet  Google Scholar 

  47. M. Sharir, A. Sheffer, J. Solymosi, Distinct distances on two lines. J. Comb. Theory Ser. A 120, 1732–1736 (2013)

    Article  MathSciNet  Google Scholar 

  48. A. Sheffer, The polynomial method, lecture notes from a course at Caltech (2015), http://www.math.caltech.edu/~2014-15/3term/ma191c-sec2/

  49. A. Sheffer, E. Szabó, J. Zahl, Point-curve incidences in the complex plane. Combinatorica 38(2), 487–499 (April 2018), arXiv:1502.0

    Article  MathSciNet  Google Scholar 

  50. A. Sheffer, J. Zahl, F. de Zeeuw, Few distinct distances implies no heavy lines or circles. Combinatorica 36(3), 349–364 (June 2016), arXiv:1308.5620

    Article  MathSciNet  Google Scholar 

  51. C.-Y. Shen, Algebraic methods in sum-product phenomena. Isr. J. Math. 188, 123–130 (2012)

    Article  MathSciNet  Google Scholar 

  52. J.H. Silverman, J. Tate, Rational Points on Elliptic Curves (Springer, Berlin, 1992)

    Book  Google Scholar 

  53. J. Solymosi, F. de Zeeuw, Incidence bounds for complex algebraic curves on cartesian products (2018) [this volume]

    Google Scholar 

  54. J. Solymosi, Bounding multiplicative energy by the sumset. Adv. Math. 222, 402–408 (2009)

    Article  MathSciNet  Google Scholar 

  55. J. Spencer, E. Szemerédi, W.T. Trotter, Unit distances in the Euclidean plane, in Graph Theory and Combinatorics, (Academic Press, Dublin, 1984), pp. 293–303

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  57. T. Tao, blog post (2012), http://terrytao.wordpress.com/2012/08/31/the-lang-weil-bound/

  58. T. Tao, Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets. Contrib. Discret. Math. 10, 22–98 (2015)

    MathSciNet  MATH  Google Scholar 

  59. P. Ungar, \(2N\) noncollinear points determine at least \(2N\) directions. J. Comb. Theory Ser. A 33, 343–347 (1982)

    Article  MathSciNet  Google Scholar 

  60. C. Valculescu, F. de Zeeuw, Distinct values of bilinear forms on algebraic curves, Contrib. Discret. Math (2015). arXiv:1403.3867

  61. H. Wang, Exposition of Elekes Szabo paper, arXiv:1512.04998 (2015)

  62. J. Zahl, A Szemerédi–Trotter type theorem in \({\mathbb{R}^4}\). Discret. Comput. Geom. 54, 513–572 (2015)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Frank de Zeeuw

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Editors and Affiliations

  1. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gergely Ambrus

  2. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  3. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Károly J. Böröczky

  4. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gábor Fejes Tóth

  5. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    János Pach

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de Zeeuw, F. (2018). A Survey of Elekes-Rónyai-Type Problems. In: Ambrus, G., Bárány, I., Böröczky, K., Fejes Tóth, G., Pach, J. (eds) New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_5

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