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 ... more >>>

We describe an explicit and simple subset of the discrete hypercube which cannot be exactly covered by fewer than exponentially many hyperplanes. The proof exploits a connection to communication complexity, and relies heavily on Razborov's lower bound for disjointness.

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