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 AC 0[⊕] 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 AC 0[⊕] circuits of linear size and deterministic AC 0[⊕] 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 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 Ω(logΘ(d) s · log 1/ε ] on the degree of ε-approximating polynomials for AC0[⊕] circuits of size s.