We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set
\(\Sigma \subset {\mathbb R}^n\) is bounded from below by
$$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\Sigma ))}{m+1} -c_2n, \end{aligned}$$
where
\(\mathrm{b}_m(\Sigma )\) is the
mth Betti number of
\(\Sigma \) with respect to “ordinary” (singular) homology and
\(c_1,\ c_2\) 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
\(\rho :\> {\mathbb R}^n \rightarrow {\mathbb R}^{n-r}\) is the projection map, then the height of any tree deciding membership in
\(\Sigma \) is bounded from below by
$$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\rho (\Sigma )))}{(m+1)^2} -\frac{c_2n}{m+1} \end{aligned}$$
for some positive constants
\(c_1,\ c_2\). We illustrate these general results by examples of lower complexity bounds for some specific computational problems.