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, for the first time, an $\Omega(n^2)$ {\em randomized} lower bound for the {\em Knapsack Problem} which was previously only known for deterministic algebraic decision trees. It is worth noting that for the languages being finite unions of hyperplanes our proof method yields also a new elementary technique for deterministic algebraic decision trees without making use of Milnor''s bound on Betti number of algebraic varieties.
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Improved lower bound on testing membership to a polyhedron by algebraic decision trees
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Lower bound on testing membership to a polyhedron by algebraic decision and computation trees
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