We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set Σ⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma \subset {\mathbb R}^n$$\end{document} is bounded from below by c1log(bm(Σ))m+1-c2n,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\Sigma ))}{m+1} -c_2n, \end{aligned}$$\end{document}where bm(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{b}_m(\Sigma )$$\end{document} is the mth Betti number of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} with respect to “ordinary” (singular) homology and c1,c2\documentclass[12pt]{minimal}
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\begin{document}$$c_1,\ c_2$$\end{document} are some (absolute) positive constants. This result complements the well-known lower bound by Yao (J Comput Syst Sci 55:36–43, 1997) for locally closed semialgebraic sets in terms of the total Borel–Moore Betti number. We also prove that if ρ:Rn→Rn-r\documentclass[12pt]{minimal}
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\begin{document}$$\rho :\> {\mathbb R}^n \rightarrow {\mathbb R}^{n-r}$$\end{document} is the projection map, then the height of any tree deciding membership in Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is bounded from below by c1log(bm(ρ(Σ)))(m+1)2-c2nm+1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\rho (\Sigma )))}{(m+1)^2} -\frac{c_2n}{m+1} \end{aligned}$$\end{document}for some positive constants c1,c2\documentclass[12pt]{minimal}
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\begin{document}$$c_1,\ c_2$$\end{document}. We illustrate these general results by examples of lower complexity bounds for some specific computational problems.
