Skip to main content
SpringerLink
Log in

The Elekes–Szabó Theorem in four dimensions

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ⊂ C be finite sets of size n. We show that F vanishes on at most O(n8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Ball, On sets defining few ordinary planes, Discrete & Computational Geometry, to appear, available at arXiv:1606.02138 (2016).

  2. M. Charalambides, Distinct distances on curves via rigidity, Discrete & Computational Geometry 51 (2014), 666–701.

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Elekes and L. Rónyai, A combinatorial problem on polynomials and rational functions, Journal of Combinatorial Theory. Series A 89 (2000), 1–20.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Harris, Algebraic Geometry: A First Course, Graduate Texts inMathematics, Vol. 133, Springer-Verlag, New York, 1992.

    Google Scholar 

  6. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York–Heidelberg, 1977.

    Google Scholar 

  7. D. Husemöller, Elliptic Curves, Graduate Texts in Mathematics, Vol. 111, Springer-Verlag, New York, 2004.

    Google Scholar 

  8. G. Károlyi, Incidence geometry in combinatorial arithmetic, Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica 52 (2009), 37–43.

    MathSciNet  MATH  Google Scholar 

  9. A. Lin, M. Makhul, H. Nassajian Mojarrad, J. Schicho, K. Swanepoel and F. de Zeeuw, On sets defining few ordinary circles, Discrete & Computational Geometry 59 (2018), 59–87.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Muntingh, Topics in Polynomial Interpolation Theory, Ph.D. dissertation, University of Oslo, 2010.

    Google Scholar 

  11. B. Murphy, O. Roche-Newton and I. Shkredov, Variations on the sum-product problem, SIAM Journal on Discrete Mathematics 29 (2015), 514–540.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Nassajian Mojarrad, T. Pham, C. Valculescu and F. de Zeeuw, Schwartz–Zippel bounds for two-dimensional products, Discrete Analysis (2017), paper no. 20.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. O. E. Raz, M. Sharir and J. Solymosi, Polynomials vanishing on grids: The Elekes–Rónyai problem revisited, American Journal of Mathematics 138 (2016), 1029–1065.

    Article  MathSciNet  MATH  Google Scholar 

  15. O. E. Raz, M. Sharir and F. de Zeeuw, Polynomials vanishing on Cartesian products: The Elekes–Szabó Theorem revisited, Duke Mathematical Journal 165 (18) (2016), 3517–3566.

    Article  MathSciNet  MATH  Google Scholar 

  16. O. Roche-Newton, A new expander and improved bounds for A(A + A), available at arXiv:1603.06827 (2016).

  17. J. Schwartz, Fast probabilistic algorithms for verification of polynomial identities, Journal of the Association for Computing Machinery 27 (1980), 701–717.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Schwartz, J. Solymosi and F. de Zeeuw, Extensions of a result of Elekes and Rónyai, Journal of Combinatorial Theory. Series A 120 (2013), 1695–1713.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 106, Springer-Verlag, Dordrecht, 2009.

    Google Scholar 

  20. J. Solymosi and F. de Zeeuw, Incidence bounds for complex algebraic curves on Cartesian products, in New Trends in Intuitive Geometry, Bolyai Society Mathematical Studies, Vol. 27, Springer, Berlin–Heidelberg, to appear.

  21. T. Tao and V. Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics, Vol. 105, Cambridge University Press, Cambridge, 2006.

    Google Scholar 

  22. F. de Zeeuw, A survey of Elekes–Rónyai-type problems, in New Trends in Intuitive Geometry, Bolyai Society Mathematical Studies, Vol. 27, Springer, Berlin–Heidelberg, to appear.

  23. R. Zippel, An explicit separation of relativised random polynomial time and relativised deterministic polynomial time, Information Processing Letters 33 (1989), 207–212.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, Tel Aviv University, Ramat Aviv, Tel Aviv, 6997801, Israel

    Orit E. Raz & Micha Sharir

  2. Mathematics Department, EPFL, CH-1015, Lausanne, Switzerland

    Frank de Zeeuw

Authors
  1. Orit E. Raz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Micha Sharir
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Frank de Zeeuw
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frank de Zeeuw.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raz, O.E., Sharir, M. & de Zeeuw, F. The Elekes–Szabó Theorem in four dimensions. Isr. J. Math. 227, 663–690 (2018). https://doi.org/10.1007/s11856-018-1728-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-018-1728-7

Search

Navigation