Influences in low-degree polynomials

Definition 3 A decision tree is a binary tree T. Each internal node of T is labeled with a variable xi and each leaf is labeled with a value −1 or 1. Given an input x ∈ {−1, 1}n, the tree is evaluated as follows. Start at the root; if this is a leaf then stop. Otherwise, query the value of the variable xi. If xi = −1 then recursively evaluate the left subtree, if xi = 1 then recursively evaluate the right subtree. The output of the tree is the value (−1 or 1) of the leaf that is reached eventually. Note that an input x deterministically determines the leaf, and thus the output, that the procedure ends up in. We say a decision tree computes p if its output equals p(x), for all x ∈ {−1, 1}n. We define D(p), the decision tree complexity of p, as the depth of an optimal (= minimal-depth) decision tree that computes p.