In this paper, we introduce and develop the method of certifying polynomials for proving AC0[] circuit lower bounds.

We use this method to show that Approximate Majority cannot be computed by AC0[] circuits of size n1+o(1). This implies a separation between the power of AC0[] circuits of near-linear size and uniform AC0[] (and even AC0 ) circuits of polynomial size. This also implies a separation between randomized AC0[] circuits of linear size and deterministic AC0[] 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 AC0 circuits of size n1+o(1). At the technical level, we show that for every AC0[] circuit C of near linear size, there is a low degree variety V over F2 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 (log(d)slog(1)) on the degree of -approximating polynomials for AC0[] circuits of size s.