Certifying polynomials for AC^0(parity) circuits, with applications

In this paper, we introduce and develop the method of certifying polynomials for proving AC^0 circuit lower bounds. We use this method to show that Approximate Majority cannot be computed by AC^0(parity) circuits of size n^{1 + o(1)}. This implies a separation between the power of AC^0(parity) circuits of near-linear size and uniform AC^0(parity) (and even AC^0) circuits of polynomial size. This also implies a separation between randomized AC^0(parity) circuits of linear size and deterministic AC^0(parity) circuits of near-linear size. Our proof using certifying polynomials extends the deterministic restrictions technique of Chaudhuri and Radhakrishnan, who showed that Approximate Majority cannot be computed by AC^0 circuits of size n^{1+o(1)}. At the technical level, we show that for every ACP circuit C of near-linear size, there is a low degree variety V over F_2 such that the restriction of C to V is constant. We also prove other results exploring various aspects of the power of certifying polynomials. In the process, we show an essentially optimal lower bound of Omega\left(\log^{\Theta(d)} s \cdot \log \frac{1}{\epsilon} \right) on the degree of \epsilon-approximating polynomials for AC^0(parity) circuits of size s.