Skip to main content
SpringerLink
Log in

Combinatorial Nullstellensatz over division rings

  • Published:
Journal of Algebraic Combinatorics Aims and scope Submit manuscript

Abstract

We extend Alon’s Combinatorial Nullstellensatz from polynomial rings over fields to polynomial rings over division rings and to rings of polynomial functions over centrally finite division algebras. We apply our results to extend classical theorems from additive number theory to the additive theory of division rings, where the size of algebraic sets is measured by their rank in the sense of Lam.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Variations of these results appeared in earlier works of Alon and his colleagues, but [2] is the seminal paper on the subject.

  2. We denote the standard non-real generators of \({\mathbb {H}}\) by \({\textrm{i}}, {\textrm{j}},{\textrm{k}}\), as opposed to the letters i, j, k which we reserve for indices.

  3. Recall that \(b^{b-a}\) denotes the conjugation \((b-a)b(b-a)^{-1}\) of b by \(b-a\).

  4. If K is of characteristic 0, then this condition is void.

  5. We note that in the literature there are alternative notations for these rings—for example in [8], for \(n=1\), the ring of polynomials in a central variable x over D is denoted by \(D_L[x]\), while the ring of polynomial functions in the variable x over D is denoted by \(D_G[x]\).

  6. This stands in stark contrast to the case of finite field extensions. For example, consider the linear polynomial \(z \in {\mathbb {C}}[z]\). Its real and imaginary components are of course not polynomial functions in z.

  7. Note that here substitution at a point in \(D^n\) is a homomorphism—unlike the situation considered in the previous sections.

References

  1. Alon, N., Füredi, Z.: Coverings of the cube by affine hyperplanes. European J. Combin. 14(2), 79–83 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N.: Combinatorial Nullstellensatz. Combin. Probab. Comput. 8(1–2), 7–29 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon, N., Nathanson, M.B., Ruzsa, I.: Adding distinct congruence classes modulo a prime. Amer. Math. Monthly 102(3), 250–255 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alon, G., Paran, E.: A central quaternionic Nullstellensatz. J. Algebra 574(15), 252–261 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alon, G., Paran, E.: A quaternionic Nullstellensatz. J. Pure Appl. Algebra 225(4), 106572 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bao, Z., Reichstein, Z.: A non-commutative Nullstellensatz. J. Algebra Appl. 22(4), 2350092 (2023)

    Article  Google Scholar 

  7. Erdős, P.: Some problems in number theory. In: Atkin, A.O.L., Birch, B.J. (eds.) Computers in Number Theory, pp. 405–414. Academic Press, New York (1971)

    Google Scholar 

  8. Gordon, B., Motzkin, T.S.: On the Zeros of polynomials over division rings. Trans. Amer. Math. Soc. 116, 218–226 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hungerford, T.W.: Algebra. Springer, Berlin (1974)

    MATH  Google Scholar 

  10. Károlyi, G.: A compactness argument in the additive theory and the polynomial method. Discrete Math. 302(1–3), 124–144 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lam, T.Y.: A general theory of Vandermonde matrices. Expo. Math. 4, 193–215 (1986)

    MathSciNet  MATH  Google Scholar 

  12. Lam, T.Y.: A First Course in Noncommutative Rings. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  13. Lam, T.Y., Leroy, A.: Wedderburn polynomials over division rings, I. J. Pure Appl. Algebra 186(1), 43–76 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lam, T.Y., Leroy, A.: Vandermonde and Wronsksian matrices over division rings. J. Algebra 119, 308–336 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lam, T.Y., Leroy, A.: Hilbert 90 theorems over division rings. Trans. Amer. Math. Soc. 345(2), 595–622 (1994)

    MathSciNet  MATH  Google Scholar 

  16. Lam, T.Y., Leroy, A., Ozturk, A.: Wedderburn polynomials over division rings, II. Contemp. Math. 456, 73–98 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Michalek, M.: A short proof of combinatorial Nullstellensatz. Amer. Math. Monthly 117(9), 821–823 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ore, O.: Theory of non-commutative polynomials. Ann. of Math. 34(3), 480–508 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  19. Silva, J.A.D., Hamidoune, Y.O.: Cyclic spaces for Grassmann derivatives and additive theory. Bull. Lond. Math. Soc. 26(2), 140–146 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Vishnoi, N.K.: An algebraic proof of Alon’s combinatorial Nullstellensatz. Congr. Numer. 152, 89–91 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Wilczynski, D.M.: On the fundamental theorem of algebra for polynomial equations over real composition algebras. J. Pure Appl. Algebra 218, 1195–1205 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author wishes to thank Francois Legrand and Shoni Gilboa for useful conversations on the topic of this paper, and the anonymous referee, for his/her careful reading and many comments which helped improve the presentation of this work.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, The Open University of Israel, 4353701, Ra’anana, Israel

    Elad Paran

Authors
  1. Elad Paran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elad Paran.

Ethics declarations

Conflict of interest

The author declares that he has no financial or non-financial interests that are directly or indirectly related to the work submitted for publication. The author declares that he has no conflict of interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paran, E. Combinatorial Nullstellensatz over division rings. J Algebr Comb 58, 895–911 (2023). https://doi.org/10.1007/s10801-023-01263-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-023-01263-1

Keywords

Search

Navigation