computational complexity

, Volume 4, Issue 4, pp 301–313

On the degree of boolean functions as real polynomials


  • Noam Nisan
    • The Hebrew University
  • Mario Szegedy
    • AT & T Bell Laboratories

DOI: 10.1007/BF01263419

Cite this article as:
Nisan, N. & Szegedy, M. Comput Complexity (1994) 4: 301. doi:10.1007/BF01263419


Every Boolean function may be represented as a real polynomial. In this paper, we characterize the degree of this polynomial in terms of certain combinatorial properties of the Boolean function.

Our first result is a tight lower bound of Ω(logn) on the degree needed to represent any Boolean function that depends onn variables.

Our second result states that for every Boolean functionf, the following measures are all polynomially related:
  • o The decision tree complexity off.

  • o The degree of the polynomial representingf.

  • o The smallest degree of a polynomialapproximating f in theLmax norm.

Key words

Approximationblock sensitivityBoolean functionsFourier degree

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© Birkhäuser Verlag 1994