1 research outputs found
An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean Function
We prove that there is a constant such that every Boolean
function of degree at most (as a polynomial over ) is a -junta, i.e. it depends on at most variables. This improves
the upper bound of Nisan and Szegedy [Computational Complexity
4 (1994)]. Our proof uses a new weighting scheme where we assign weights to
variables based on the highest degree monomial they appear on.
The bound of is tight up to the constant as a lower bound of
is achieved by a read-once decision tree of depth . We slightly
improve the lower bound by constructing, for each positive integer , a
function of degree with relevant variables. A similar
construction was independently observed by Shinkar and Tal.Comment: 6 page