Algebraic Decision Trees and Euler Characteristics

Abstract For any set S ⊆ R n , 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 log 3 ¦χ(S)¦ .

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