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·2d-junta, i.e., it depends on at most C·2d variables. This improves the d·2d-1 upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)]. The bound of C·2d is tight up to the constant C, since a read-...More