Degree complexity of Boolean functions and its applications to relativized separations

It is shown that a simple function in AC/sup 0/, OR of square root n disjoint ANDs, cannot be computed by decision trees of depth log/sup O(1)/n where each node asks whether or not p(x/sub 1/, . . .,x/sub n/)=0 for some polynomial p of degree log/sup O(1)/n. This is in contrast to recent results that every function in AC/sup 0/ 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(x/sub 1/, . . .,x/sub n/)>0. The proofs are based on simple algebraic arguments that also provide alternative proofs for some known results.<>