Algebraic decision trees and Euler characteristics

For any set S is contained as a subset within Rn, let χ(S) denote its Euler characteristic. In this paper, we show that any algebraic computation tree or fixed-degree algebraic decision tree must have height Ω(log|χ(S)|-cn) for deciding the membership question of a compact semi-algebraic set S. This extends a result in Bjorner et al. (1992), where it was shown that any linear decision tree for deciding the membership question of a closed polyhedron S must have height greater than or equal to log3|χ(S)|.