On the degree of boolean functions as real polynomials

Noam Nisan*, Mario Szegedy

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

273 Scopus citations

Abstract

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(log n) on the degree needed to represent any Boolean function that depends on n variables. Our second result states that for every Boolean function f, the following measures are all polynomially related:o The decision tree complexity of f. o The degree of the polynomial representing f. o The smallest degree of a polynomial approximating f in the Lmax norm.

Original languageAmerican English
Pages (from-to)301-313
Number of pages13
JournalComputational Complexity
Volume4
Issue number4
DOIs
StatePublished - Dec 1994

Keywords

  • Approximation
  • Boolean functions
  • Fourier degree
  • Subject classifications: 68Q05, 68Q99
  • block sensitivity

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