Advertisement

Combinatorica

, Volume 32, Issue 5, pp 589–605 | Cite as

Alon’s Nullstellensatz for multisets

  • Géza KósEmail author
  • Lajos Rónyai
Article

Abstract

Alon’s combinatorial Nullstellensatz (Theorem 1.1 from [2]) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let \(\mathbb{F}\) be a field, S 1, S 2,..., S n be finite nonempty subsets of \(\mathbb{F}\). Alon’s theorem is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing on the set \(S = S_1 \times S_2 \times \ldots \times S_n \subseteq \mathbb{F}^n\). From this Alon deduces a simple and amazingly widely applicable nonvanishing criterion (Theorem 1.2 in [2]). It provides a sufficient condition for a polynomial f(x 1,..., x n) which guarantees that f is not identically zero on the set S. In this paper we extend these two results from sets of points to multisets. We give two different proofs of the generalized nonvanishing theorem.We extend some of the known applications of the original nonvanishing theorem to a setting allowing multiplicities, including the theorem of Alon and Füredi on the hyperplane coverings of discrete cubes.

Mathematics Subject Classification (2000)

05-XX 05E40 12D10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    W. W. Adams, P. Loustaunau: An introduction to Gröbner bases, American Mathematical Society, 1994.Google Scholar
  2. [2]
    N. Alon: Combinatorial Nullstellensatz, Combinatorics, Probability and Computing 8 (1999), 7–29.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    N. Alon, Z. Füredi: Covering the cube by affine hyperplanes, European J. Combinatorics 14 (1993), 79–83.zbMATHCrossRefGoogle Scholar
  4. [4]
    M. F. Atiyah, I. G. Macdonald: Introduction to commutative algebra, Addison-Wesley, 1969.Google Scholar
  5. [5]
    S. Ball, O. Serra: Punctured Combinatorial Nullstellensätze, Combinatorica 29 (2009), 511–522.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    C. de Boor: Divided differences, Surveys in Approximation Theory 1 (2005), 46–69.MathSciNetzbMATHGoogle Scholar
  7. [7]
    B. Buchberger: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Doctoral thesis, University of Innsbruck, 1965. English Translation: An algorithm for finding the basis elements in the residue class ring modulo a zero dimensional polynomial ideal. Journal of Symbolic Computation Special Issue on Logic, Mathematics, and Computer Science: Interactions. 41 (2006), 475–511.Google Scholar
  8. [8]
    M. Cámara, A. Lladó, J. Moragas: On a conjecture of Graham and Häggkvist with the polynomial method, European Journal of Combinatorics 30 (2009), 1585–1592.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    A. M. Cohen, H. Cuypers, H. Sterk: Some Tapas of Computer Algebra, Springer-Verlag, 1999.Google Scholar
  10. [10]
    Z. Dvir, S. Kopparty, S. Saraf, M. Sudan: Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers, arXiv:0901.2529v2Google Scholar
  11. [11]
    S. Eliahou, M. Kervaire: Sumsets in vector spaces over finite fields, Journal of Number Theory 71 (1988), 12–39.MathSciNetCrossRefGoogle Scholar
  12. [12]
    S. Eliahou, M. Kervaire: Old and new formulas for the Hopf-Stiefel and related functions, Expositiones Mathematicae 23 (2005), 127–145.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    B. Felszeghy: On the solvability of some special equations over finite fields, Publicationes Mathematicae Debrecen 68 (2006), 15–23.MathSciNetzbMATHGoogle Scholar
  14. [14]
    B. Green, T. Tao: The distribution of polynomials over finite fields, with applications to the Gowers norms, Contributions to Discrete Mathematics 4 (2009), 1–36.MathSciNetzbMATHGoogle Scholar
  15. [15]
    Gy. Károlyi: Cauchy-Davenport theorem in group extensions, L’Enseignement Mathématique 51 (2005), 239–254.zbMATHGoogle Scholar
  16. [16]
    Gy. Károlyi: Restricted set addition: the exceptional case of the Erdős-Heilbronn conjecture, Journal of Combinatorial Theory, Ser. A 116 (2009), 741–746.zbMATHCrossRefGoogle Scholar
  17. [17]
    H. Pan, Z-W. Sun: A new extension of the Erdős-Heilbronn conjecture, Journal of Combinatorial Theory, Ser. A 116 (2009), 1374–1381.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Z-W. Sun: On value sets of polynomials over a field, Finite Fields and Applications 14 (2008), 470–481.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Computer and Automation Research InstuteHungarian Academy of SciencesBudapestHungary
  2. 2.Department of AnalysisEötvös Loránd UniversityBudapestHungary
  3. 3.Department of AlgebraBudapest University of Technology and EconomicsBudapestHungary

Personalised recommendations