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(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 L_max norm.