Decision Tree Complexity and Betti Numbers

We show that any algebraic computation tree or any fixed-degree algebraic tree for solving the membership question of a compact setS?Rnmust have height greater than?(log(si(S)))?cnfor eachi, wheresi(S) is theith Betti number. This generalizes a well-known result by Ben-Or who proved this lower bound for the casei=0, and a recent result by Bjorner and Lovasz who proved this lower bound for allifor linear decision trees.

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