An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean function

We prove that there is a constant C ≤ 6.614 such that every Boolean function of degree at most d (as a polynomial over ℝ) is a C·2^d-junta, i.e., it depends on at most C·2^d variables. This improves the d·2^d-1 upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)]. The bound of C·2^d is tight up to the constant C, since a read-once decision tree of depth d depends on all 2^d - 1 variables. We slightly improve this lower bound by constructing, for each positive integer d, a function of degree d with 3·2^d-1 - 2 relevant variables. A similar construction was independently observed by Shinkar and Tal.