Algebras and Representation Theory

, Volume 21, Issue 5, pp 1017–1021 | Cite as

On Ranks of Polynomials

  • David Kazhdan
  • Tamar Ziegler


Let V be a vector space over a field k, P : Vk, d ≥ 3. We show the existence of a function C(r, d) such that rank(P) ≤ C(r, d) for any field k, char(k) > d, a finite-dimensional k-vector space V and a polynomial P : Vk of degree d such that rank(P/t) ≤ r for all tV − 0. Our proof of this theorem is based on the application of results on Gowers norms for finite fields k. We don’t know a direct proof even in the case when k = .

Mathematics Subject Classification (2010)

11B30 14J99 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bhowmick, A, Lovett, S: Bias vs structure of polynomials in large fields, and applications in effective. arXiv:1506.02047
  2. 2.
    Derksen, H, Eggermont, R., Snowden, A.: Topological noetherianity for cubic polynomials. arXiv:1701.01849
  3. 3.
    Gowers, T.: A new proof of szemerédi’s theorem. Geom. Funct. Anal. 11:465–588 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hartshorne, R.: Algebraic Geometry Graduate Texts in Mathematics, No. 52. Springer, Heidelberg (1977)Google Scholar
  5. 5.
    Kazhdan, D., Ziegler, T.: Approximate cohomology. Sel. Math. New Ser (2017)Google Scholar
  6. 6.
    Markel, D.: Model theory:an introduction graduate texts of mathematics. SpringerGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsGivaat Ram The Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations