On the degree of boolean functions as real polynomials
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 wordsApproximation block sensitivity Boolean functions Fourier degree
Subject classifications68Q05 68Q99