On the degree of boolean functions as real polynomials

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 epends on n variables. Our second result states that for every boolean function f the following measures are all polynomi-ally related: The decision tree complexity of f. The degree of the polynomial representing f. The smallest degree of a polynomial approximating f in the Lmax norm.