Dual Lower Bounds for Approximate Degree and Markov-Bernstein Inequalities

The ε-approximate degree of a Boolean function f : {−1, 1} → {−1, 1} is the minimum degree of a real polynomial that approximates f to within ε in the `∞ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the ε-approximate degree of the two-level AND-OR tree for any constant ε > 0. We show that this quantity is Θ( √ n), closing a line of incrementally larger lower bounds [3, 12, 25, 36, 38]. The same lower bound was recently obtained independently by Sherstov using related techniques [31]. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Špalek [40]. Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.