Degree complexity of Boolean functions and its applications to relativized separations

It is shown that a simple function in AC⁰, OR of √n disjoint ANDs, cannot be computed by decision trees of depth log^O(1)n where each node asks whether or not p(x₁, . . .,x_n)=0 for some polynomial p of degree log^O(1)n. This is in contrast to recent results that every function in AC⁰ can be computed probabilistically by just one such query and can be deterministically computed by such decision trees if each node asks whether or not p(x₁, . . .,x_n )>0. The proofs are based on simple algebraic arguments that also provide alternative proofs for some known results