An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean function


We prove that there is a constant C ≤ 6.614 such that every Boolean function of degree at most d (as a polynomial over ℝ) is a C·2d-junta, i.e., it depends on at most C·2d variables. This improves the d·2d-1 upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)].

The bound of C·2d is tight up to the constant C, since a read-once decision tree of depth d depends on all 2d - 1 variables. We slightly improve this lower bound by constructing, for each positive integer d, a function of degree d with 3·2d-1 - 2 relevant variables. A similar construction was independently observed by Shinkar and Tal.

This is a preview of subscription content, access via your institution.


  1. [1]

    H. Buhrman and R. de Wolf: Complexity measures and decision tree complexity: a survey, Theoretical Computer Science 288 (2002), no. 1, 21–43.

    MathSciNet  Article  Google Scholar 

  2. [2]

    Y. Filmus and F. Ihringer: Boolean constant degree functions on the slice are juntas, arXiv:1801.06338 (2018).

    Google Scholar 

  3. [3]

    P. Hatami, R. Kulkarni and D. Pankratov: Variations on the sensitivity conjecture, Theory of Computing Library, Graduate Surveys 4 (2011), 1–27.

    Google Scholar 

  4. [4]

    M. L. Minsky and S. A. Papert: Perceptrons, MIT press (1988).

    Google Scholar 

  5. [5]

    N. Nisan and M. Szegedy: On the degree of Boolean functions as real polynomials, Computational complexity 4 (1994), 301–313.

    MathSciNet  Article  Google Scholar 

  6. [6]

    R. O’DONNELL: Analysis of Boolean functions, Cambridge University Press, 2014.

    Google Scholar 

  7. [7]

    I. Shinkar and A. Tal: Private communication (2017).

    Google Scholar 

  8. [8]

    A. Tal: Properties and applications of Boolean function composition, Proceedings of the 4th conference on Innovations in Theoretical Computer Science, ACM, 2013, 441–454.

    Google Scholar 

  9. [9]

    J. Wellens: Private Communication (2019).

    Google Scholar 

Download references


We thank Avishay Tal for helpful discussions, and for sharing his python code for exhaustive search of Boolean functions on few variables. We also thank Yuval Filmus for pointing out the implications for Boolean functions on the slice, and Jake Lee Wellens for identifying a technical error in Proposition 2.1 from a previous version of this paper. We finally thank the referees for their detailed and insightful comments.

Author information



Corresponding author

Correspondence to John Chiarelli.

Additional information

Supported in part by the Simons Foundation under award 332622.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chiarelli, J., Hatami, P. & Saks, M. An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean function. Combinatorica 40, 237–244 (2020).

Download citation

Mathematics Subject Classification (2010)

  • 05D05
  • 06E30
  • 26C05