A nullstellensatz for sequences over \(\mathbb{F}_p \)

Abstract

Let p be a prime and let A = (a 1,...,a ) be a sequence of nonzero elements in \(\mathbb{F}_p \). In this paper, we study the set of all 0–1 solutions to the equation \(a_1 x_1 + \cdots + a_\ell x_\ell = 0\) We prove that whenever p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for <p. The critical case =p is of particular interest. In this context, we prove that whenever =p and A is nonconstant, the above equation has at least p−1 minimal 0–1 solutions, thus refining a theorem of Olson. The subcritical case =p−1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.

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References

  1. [1]

    N. Alon: Combinatorial Nullstellensatz, Combin. Probab. Comput. 8 (1999), 7–29.

    Article  MATH  MathSciNet  Google Scholar 

  2. [2]

    E. Balandraud: An addition theorem and maximal zero-sum free sets in ℤ/pℤ, Israel J. Math. 188 (2012), 405–429.

    Article  MATH  MathSciNet  Google Scholar 

  3. [3]

    A.-L. Cauchy: Recherches sur les nombres, J. Ecole Polytech. 9 (1813), 99–116.

    Google Scholar 

  4. [4]

    H. Davenport: On the addition of residue classes, J. Lond. Math. Soc. 10 (1935), 30–32.

    Google Scholar 

  5. [5]

    H. Davenport: A historical note, J. Lond. Math. Soc. 22 (1947), 100–101.

    Article  MATH  MathSciNet  Google Scholar 

  6. [6]

    P. Erdös, A. Ginzburg and A. Ziv: Theorem in the additive number theory, Bull. Research Council Israel 10 (1961), 41–43.

    Google Scholar 

  7. [7]

    A. Geroldinger: Additive group theory and non-unique factorizations, in A. Geroldinger and I. Ruzsa, Combinatorial Number Theory and Additive Group Theory, Advanced Courses in Mathematics, CRM Barcelona Birkh auser (2009), 1–86.

    Google Scholar 

  8. [8]

    A. Geroldinger and F. Halter-Koch: Non-unique factorizations. Algebraic, combinatorial and analytic theory, Pure and Applied Mathematics 278, Chapman & Hall/CRC (2006).

  9. [9]

    J. E. Olson: A problem of Erdos on abelian groups, Combinatorica 7 (1987), 285–289.

    Article  MATH  MathSciNet  Google Scholar 

  10. [10]

    T. Tao and V. H. Vu: Additive Combinatorics, Cambridge Studies in Advanced Mathematics 105 (2006), Cambridge Press University.

  11. [11]

    G. Vosper: The critical pairs of subsets of a group of prime order, J. Lond. Math. Soc. 31 (1956), 200–205.

    Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    G. Vosper: Addendum to “The critical pairs of subsets of a group of prime order”, J. Lond. Math. Soc. 31 (1956), 280–282

    Article  MathSciNet  Google Scholar 

Download references

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Correspondence to Éric Balandraud or Benjamin Girard.

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Research supported by the French ANR Project “CAESAR” No. ANR-12-BS01-0011

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Balandraud, É., Girard, B. A nullstellensatz for sequences over \(\mathbb{F}_p \) . Combinatorica 34, 657–688 (2014). https://doi.org/10.1007/s00493-011-2961-4

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Mathematics Subject Classification (2010)

  • 11D04
  • 11T06
  • 11D45
  • 11P70