Alon’s Nullstellensatz for multisets

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.

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References

  1. [1]

    W. W. Adams, P. Loustaunau: An introduction to Gröbner bases, American Mathematical Society, 1994.

  2. [2]

    N. Alon: Combinatorial Nullstellensatz, Combinatorics, Probability and Computing 8 (1999), 7–29.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    N. Alon, Z. Füredi: Covering the cube by affine hyperplanes, European J. Combinatorics 14 (1993), 79–83.

    MATH  Article  Google Scholar 

  4. [4]

    M. F. Atiyah, I. G. Macdonald: Introduction to commutative algebra, Addison-Wesley, 1969.

  5. [5]

    S. Ball, O. Serra: Punctured Combinatorial Nullstellensätze, Combinatorica 29 (2009), 511–522.

    MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    C. de Boor: Divided differences, Surveys in Approximation Theory 1 (2005), 46–69.

    MathSciNet  MATH  Google 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.

  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.

    MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    A. M. Cohen, H. Cuypers, H. Sterk: Some Tapas of Computer Algebra, Springer-Verlag, 1999.

  10. [10]

    Z. Dvir, S. Kopparty, S. Saraf, M. Sudan: Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers, arXiv:0901.2529v2

  11. [11]

    S. Eliahou, M. Kervaire: Sumsets in vector spaces over finite fields, Journal of Number Theory 71 (1988), 12–39.

    MathSciNet  Article  Google Scholar 

  12. [12]

    S. Eliahou, M. Kervaire: Old and new formulas for the Hopf-Stiefel and related functions, Expositiones Mathematicae 23 (2005), 127–145.

    MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    B. Felszeghy: On the solvability of some special equations over finite fields, Publicationes Mathematicae Debrecen 68 (2006), 15–23.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google Scholar 

  15. [15]

    Gy. Károlyi: Cauchy-Davenport theorem in group extensions, L’Enseignement Mathématique 51 (2005), 239–254.

    MATH  Google 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.

    MATH  Article  Google 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.

    MathSciNet  MATH  Article  Google Scholar 

  18. [18]

    Z-W. Sun: On value sets of polynomials over a field, Finite Fields and Applications 14 (2008), 470–481.

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Géza Kós.

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Research supported in part by OTKA grants NK 105645, K 77476, and K 77778.

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Kós, G., Rónyai, L. Alon’s Nullstellensatz for multisets. Combinatorica 32, 589–605 (2012). https://doi.org/10.1007/s00493-012-2758-0

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Mathematics Subject Classification (2000)

  • 05-XX
  • 05E40
  • 12D10