Tight Bound on the Number of Relevant Variables in a Bounded degree Boolean function.
John Chiarelli, Pooya Hatami, Michael Saks · arXiv (Cornell University) · 2018
In this paper, we prove that a degree $d$ Boolean function depends on at most $C\cdot 2^d$ variables for some $C<22$, i.e. it is a $C\cdot 2^d$-junta. This improves the $d\cdot 2^{d-1}$ upper bound of Nisan and Szegedy [NS94]. Our proof uses a new weighting scheme where we assign weights to variables based on the highest degree monomial they appear on. We note that a bound of $C\cdot 2^d$ is essentially tight. A lower bound of $2^d-1$ can easily be achieved by a read-once decision tree of depth $d$ (see [O'Donnell 14], Exercise 3.24). We slightly improve this bound by constructing a Boolean function of degree $d$ with $3\cdot 2^{d-1}-2$ relevant variables. This construction was independently observed by Avishay Tal, but has not appeared in a publication before [Tal 17].