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 Σ ⊂ Rn is bounded from below by c1 log(bm(Σ)) m+ 1 − c2n, where bm(Σ) is the m-th Betti number of Σ with respect to “ordinary” (singular) homology, and c1, c2 are some (absolute) positive constants. This result complements the well known lower bound by Yao [7] for locally closed semialgebraic sets in terms of the total Borel-Moore Betti number. We also prove that if ρ : Rn → Rn−r is the projection map, then the height of any tree deciding membership in Σ is bounded from below by c1 log(bm(ρ(Σ))) (m+ 1)2 − c2n m+ 1 for some positive constants c1, c2. We illustrate these general results by examples of lower complexity bounds for some specific computational problems.