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

Warning’s Second Theorem with restricted variables

  • Pete L. Clark
  • Aden Forrow
  • John R. Schmitt
Original Paper


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)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    N. Alon and Z. Füredi: Covering the cube by affne hyperplanes, Eur. J. Comb. 14 (1993), 79–83.CrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. Brink: Chevalley’s theorem with restricted variables, Combinatorica 31 (2011), 127–130.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    L. Carlitz: A Characterization of Algebraic Number Fields with Class Number Two, Proc. AMS 11 (1960), 391–392.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    P. L. Clark: The Combinatorial Nullstellensätze Revisited, Electronic Journal of Combinatorics 21 (2014), Paper #P4.15.zbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L. E. Dickson: On the representation of numbers by modular forms, Bull. Amer. Math. Soc. 15 (1909), 338–347.MathSciNetCrossRefzbMATHGoogle 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.zbMATHGoogle Scholar
  15. [15]
    P. Erdős, A. Ginzburg and A. Ziv: Theorem in the additive number theory, Bull. Research Council Israel 10F (1961), 41–43.zbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    N. M. Katz: On a theorem of Ax, Amer. J. Math. 93 (1971), 485–499.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Lasoń: A generalization of combinatorial Nullstellensatz, Electron. J. Combin. 17 (2010), Note 32.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  22. [22]
    J. E. Olson: A combinatorial problem on finite Abelian groups, I. J. Number Theory 1 (1969), 8–10.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J. E. Olson: A combinatorial problem on finite Abelian groups, II, J. Number Theory 1 (1969), 195–199.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    S. H. Schanuel: An extension of Chevalley’s theorem to congruences modulo prime powers, J. Number Theory 6 (1974), 284–290.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    U. Schauz: Algebraically solvable problems: describing polynomials as equivalent to explicit solutions, Electron. J. Combin. 15 (2008), Research Paper 10.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    R. J. Valenza: Elasticity of factorizations in number fields, J. Number Theory 36 (1990), 212–218.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    E. Warning: Bemerkung zur vorstehenden Arbeit von Herrn Chevalley, Abh. Math. Sem. Hamburg 11 (1935), 76–83.MathSciNetCrossRefzbMATHGoogle 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