Improved lower bound on testing membership to a polyhedron by algebraic decision trees

We introduce a new method of proving lower bounds on the depth of algebraic decision trees of degree d and apply it to prove a lower bound Ω(log N) for testing membership to an n-dimensional convex polyhedron having N faces of all dimensions, provided that N>(nd)^Ω(n). This weakens considerably the restriction on N previously imposed by the authors and opens a possibility to apply the bound to some naturally appearing polyhedra