Combinatorica

, Volume 32, Issue 5, pp 537–571 | Cite as

How to find groups? (and how to use them in Erdős geometry?)

Article

Abstract

Geometric questions which involve Euclidean distances often lead to polynomial relations of type F(x, y, z)=0 for some F ∈ ℝ[x, y, z]. Several problems of Combinatorial Geometry can be reduced to studying such polynomials which have many zeroes on n×n×n Cartesian products. The special case when the relation F = 0 can be re-written as z = f(x, y), for a polynomial or rational function f ∈ ℝ(x, y), was considered in [8]. Our main goal is to extend the results found there to full generality (and also to show some geometric applications, e.g. one on “circle grids”).

The main result of our paper concerns low-degree algebraic sets F which contain “too many” points of a (large) n×n×n Cartesian product. Then we can conclude that, in a neighborhood of almost any point, the set F must have a very special (and very simple) form. More precisely, then either F is a cylinder over some curve, or we find a group behind the scene: F must be the image of the graph of the multiplication function of an appropriate algebraic group (see Theorem 3 for the 3D special case and Theorem 27 in full generality).

Mathematics Subject Classification (2000)

05A16 14N10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Brass and C. Knauer: On counting point-hyperplane incidences, Comput. Geom. Theory Appl. 25(12) (2003), 13–20.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir: A singly-exponential stratification scheme for real semi-algebraic varieties and its applications, Theoretical Computer Science 84 (1991), 77–105.MATHCrossRefGoogle Scholar
  3. [3]
    Gy. Elekes: Circle grids and bipartite graphs of distances, Combinatorica 15 (1995), 167–174.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Gy. Elekes: On the number of sums and products, Acta Arithmetica LXXXI.4 (1997), 365–367.MathSciNetGoogle Scholar
  5. [5]
    Gy. Elekes. Sums versus products in number theory, algebra and Erdős geometry, Paul Erdős and his mathematics, II (Budapest, 1999), Bolyai Soc. Math. Stud. 11, János Bolyai Math. Soc., Budapest, 2002, 241–290.Google Scholar
  6. [6]
    P. Erdős, L. Lovász, and K. Vesztergombi: On graphs of large distances, Discrete and Computational Geometry 4 (1989), 541–549.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Gy. Elekes, M. B. Nathanson, and I. Z. Ruzsa: Convexity and sumsets, Journal of Number Theory 83 (1999), 194–201.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Gy. Elekes and L. Rónyai: A combinatorial problem on polynomials and rational functions, Journal of Combinatorial Theory, series A 89 (2000), 1–20.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    M. Gromov: Groups of polynomial growth and expanding maps, Publ. Math., Inst. Hautes Étud. Sci. 53 (1981), 53–78.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    J. Harris: Algebraic geometry: A first course, Springer-Verlag, New York, 1992.MATHGoogle Scholar
  11. [11]
    F. Hirzebruch: Singularities of algebraic surfaces and characteristic numbers, Contemp. Math. 58 (1986), 141–155.MathSciNetCrossRefGoogle Scholar
  12. [12]
    E. Hrushovski: Contributions to stable model theory, PhD thesis, University of California, Berkeley, 1986.Google Scholar
  13. [13]
    N. Jacobson: Lectures in abstract algebra, III. Theory of fields and Galois theory, University Series in Higher Mathematics, Van Nostrand Reinhold Co., 1964.Google Scholar
  14. [14]
    J. Kollár: Rational Curves on Algebraic Varieties, Springer, Berlin, 1996.Google Scholar
  15. [15]
    J. Matoušek: Lectures on Discrete Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 2002.MATHCrossRefGoogle Scholar
  16. [16]
    G. Megyesi and E. Szabó: On the tacnodes of configurations of conics in the projective plane, Mathematische Annalen 305 (1996), 693–703.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    A. L. Onishchik and E. B. Vinberg: Lie groups and algebraic groups, Springer, Berlin, 1990.MATHCrossRefGoogle Scholar
  18. [18]
    A. Pillay: Geometric Stability Theory, volume 32 of Oxford Logic Guides, Clarendon Press, Oxford, 1996.MATHGoogle Scholar
  19. [19]
    J. Pach and M. Sharir: On the number of incidences between points and curves, Combinatorics, Probability and Computing 7 (1998), 121–127.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    E. Szemerédi and W. T. Trotter Jr.: Extremal problems in Discrete Geometry, Combinatorica 3(3–4) (1983), 381–392.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    A. Weil: On algebraic groups and homogeneous spaces, American Journal of Mathematics 77 (1955), 493–512.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    A. Weil: On algebraic groups of transformations, American Journal of Mathematics 77 (1955), 355–391.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Mathematical Institute of Eötvös UniversityBudapestHungary
  2. 2.Alfréd Rényi Mathematical InstituteBudapestHungary

Personalised recommendations