Lower Bound on Testing Membership to a Polyhedron by Algebraic Decision and Computation Trees

We introduce a new method of proving lower bounds on the depth of algebraic d-degree decision (resp. computation) trees and apply it to prove a lower bound ~2 (log N) (resp. f2 (log N/log log N)) for testing membership to an n-dimensional convex polyhedron having N faces of all dimensions, provided that N > ( n d ) ~( ' ) (resp. N > nU<n)). This bound apparently does not follow from the methods developed by Ben-Or, Bj~Srner, Lovasz, and Yao [1], [4], [24] because topological invariants used in these methods become trivial for convex polyhedra.