An exponential lower bound on the size of algebraic decision trees for max

We prove an exponential lower bound on the size of any fixed degree algebraic decision tree for solving MAX, the problem of finding the maximum of n real numbers. This complements the n - 1 lower bound of [Rabin (1972)] on the depth of algebraic decision trees for this problem. The proof in fact gives an exponential lower bound on the size of the polyhedral decision problem MAX= for testing whether the j-th number is the maximum among a list of n real numbers. Previously, except for linear decision trees, no nontrivial lower bounds on the size of algebraic decision trees for any familiar problems are known. We also establish an interesting connection between our lower bound and the maximum number of minimal cutsets for any rank-d hypergraph on n vertices.