Combinatorica

, Volume 37, Issue 3, pp 397–417 | Cite as

Warning’s Second Theorem with restricted variables

Original Paper
  • 65 Downloads

Abstract

We present a restricted variable generalization of Warning’s Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Schauz-Brink’s restricted variable generalization of Chevalley’s Theorem (a result giving conditions for a low degree polynomial system not to have exactly one solution). Just as Warning’s Second Theorem implies Chevalley’s Theorem, our result implies Schauz-Brink’s Theorem. We include several combinatorial applications, enough to show that we have a general tool for obtaining quantitative refinements of combinatorial existence theorems.

Let q = p be a power of a prime number p, and let F q be “the” finite field of order q.

For a 1,...,a n , N∈Z+, we denote by m(a 1,...,a n ;N)∈Z+ a certain combinatorial quantity defined and computed in Section 2.1.

Mathematics Subject Classification (2000)

11T99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Alon, D. Kleitman, R. Lipton, R. Meshulam, M. Rabin and J. Spencer: Set systems with no union of cardinality 0 modulo m, Graphs Combin. 7 (1991), 97–99.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    N. Alon and Z. Füredi: Covering the cube by affne hyperplanes, Eur. J. Comb. 14 (1993), 79–83.CrossRefMATHGoogle Scholar
  3. [3]
    N. Alon: Combinatorial Nullstellensatz, Recent trends in combinatorics (Mátraháza, 1995). Combin. Probab. Comput. 8 (1999), 7–29.MathSciNetCrossRefGoogle Scholar
  4. [4]
    J. Ax: Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    C. Bailey and R. B. Richter: Sum zero (mod n), size n subsets of integers, Amer. Math. Monthly 96 (1989), 240–242.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    D. Brink: Chevalley’s theorem with restricted variables, Combinatorica 31 (2011), 127–130.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    L. Carlitz: A Characterization of Algebraic Number Fields with Class Number Two, Proc. AMS 11 (1960), 391–392.MathSciNetMATHGoogle Scholar
  8. [8]
    G. J. Chang, S.-H. Chen., Y. Qu, G. Wang and H. Zhang: On the number of subsequences with a given sum in a finite abelian group, Electron. J. Combin. 18 (2011), Paper 133.MathSciNetMATHGoogle Scholar
  9. [9]
    A. Chattopadhyay, N. Goyal, P. Pudlák and D. Thérien: Lower bounds for circuits with MODm gates, Proc. 47th Annual Symp. on Foundations of Computer Science, IEEE 2006, 709–718.Google Scholar
  10. [10]
    C. Chevalley: Démonstration d’une hypothèse de M. Artin, Abh. Math. Sem. Univ. Hamburg 11 (1935), 73–75.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    P. L. Clark: The Combinatorial Nullstellensätze Revisited, Electronic Journal of Combinatorics 21 (2014), Paper #P4.15.MATHGoogle Scholar
  12. [12]
    S. Das Adhikari, D. J. Grynkiewicz and Z.-W. Sun: On weighted zero-sum sequences, Adv. in Appl. Math. 48 (2012), 506–527.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    L. E. Dickson: On the representation of numbers by modular forms, Bull. Amer. Math. Soc. 15 (1909), 338–347.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    P. van Emde Boas and D. Kruyswijk: A combinatorial problem on finite abelian groups, III, Report ZW-1969-008, Math. Centre, Amsterdam, 1969.MATHGoogle Scholar
  15. [15]
    P. Erdős, A. Ginzburg and A. Ziv: Theorem in the additive number theory, Bull. Research Council Israel 10F (1961), 41–43.MATHGoogle Scholar
  16. [16]
    H. Esnault: Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. Math. 151 (2003), 187–191.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    D. R. Heath-Brown: On Chevalley-Warning theorems, Uspekhi Mat. Nauk 66 (2011), 223–232; translation in: Russian Math. Surveys 66 (2011), 427–436.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    N. M. Katz: On a theorem of Ax, Amer. J. Math. 93 (1971), 485–499.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    R. N. Karasev and F. V. Petrov: Partitions of nonzero elements of a finite field into pairs, Israel J. Math. 192 (2012), 143–156.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    M. Lasoń: A generalization of combinatorial Nullstellensatz, Electron. J. Combin. 17 (2010), Note 32.MathSciNetMATHGoogle Scholar
  21. [21]
    D. G. Mead and W. Narkiewicz: The capacity of C 5 and free sets in C m 2, Proc. Amer. Math. Soc. 84 (1982), 308–310.MathSciNetMATHGoogle Scholar
  22. [22]
    J. E. Olson: A combinatorial problem on finite Abelian groups, I. J. Number Theory 1 (1969), 8–10.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    J. E. Olson: A combinatorial problem on finite Abelian groups, II, J. Number Theory 1 (1969), 195–199.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    S. H. Schanuel: An extension of Chevalley’s theorem to congruences modulo prime powers, J. Number Theory 6 (1974), 284–290.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    U. Schauz: Algebraically solvable problems: describing polynomials as equivalent to explicit solutions, Electron. J. Combin. 15 (2008), Research Paper 10.MathSciNetMATHGoogle Scholar
  26. [26]
    C. C. Tsen: Divisionsalgebren über Funktionenkörpern, Nachr. Ges. Wiss. Göttingen (1933), 335–339.Google Scholar
  27. [27]
    G. Troi and U. Zannier: On a theorem of J. E. Olson and an application (vanishing sums in finite abelian p-groups), Finite Fields Appl. 3 (1997), 378–384.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    R. J. Valenza: Elasticity of factorizations in number fields, J. Number Theory 36 (1990), 212–218.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    E. Warning: Bemerkung zur vorstehenden Arbeit von Herrn Chevalley, Abh. Math. Sem. Hamburg 11 (1935), 76–83.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    R. M. Wilson: Some applications of polynomials in combinatorics, EPM lectures, May, 2006.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of MathematicsMiddleburg CollegeMiddleburgUSA

Personalised recommendations