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On topological lower bounds for algebraic computation trees

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of the set, divided by m+1. This result complements the well known lower bound by Yao for locally closed semialgebraic sets in terms of the total Borel-Moore Betti number. We also prove that the height is bounded from below by the logarithm of m-th Betti number of a projection of the set onto a coordinate subspace, divided by (m+1)^2. We illustrate these general results by examples of lower complexity bounds for some specific computational problems.