We describe a new method for proving lower bounds for algebraic computation trees. We prove, for the first time, that the minimum depth for arbitrary computation trees for the problem of testing the membership to a polygon with $N$ nodes is $\Omega(\log N)$. Moreover, we prove that the corresponding lower bound for the randomized computation trees matches the above bound. Finally, we prove that for the algebraic exp-log computation trees (cf. [GSY 93]), the minimum depth is $\Omega(\sqrt{\,\log N})$. We generalize the last result to the multidimensional case, showing that if an exp-log computation tree tests a membership to a semialgebraic set with a sum of Betti numbers $M$, then the depth of a tree is at least $\Omega(\sqrt{\,\log M})$