We extend the lower bounds on the depth of algebraic decision trees to the case of {\em randomized} algebraic decision trees (with two-sided error) for languages being finite unions of hyperplanes and the intersections of halfspaces, solving a long standing open problem. As an application, among other things, we derive, ... more >>>

We survey some of the recent results on the complexity of recognizing n-dimensional linear arrangements and convex polyhedra by randomized algebraic decision trees. We give also a number of concrete applications of these results. In particular, we derive first nontrivial, in fact quadratic, randomized lower bounds on the problems like ... more >>>