Lower Bounds for Algebraic Computation Trees of Functions with Finite Domains

Let $W \subseteq R^n $ be scale-invariant and rationally dispersed, i.e., $\tilde x \in W$ implies $\lambda \tilde x \in W$ for all real $\lambda > 0$; and the rationale are dense in both W and $R^n - W$. It is shown that, in the algebraic computation tree model, the problem of deciding whether an input $\tilde x \in R^n $ with integer coordinates is a member of W has complexity $\Omega (\log _2 \hat \beta (W) - 2n)$, where $\hat \beta (W)$ is the number of connected components of W that are not of measure 0. This theorem can be used to prove tight lower bounds for the integral-constrained form of many basic problems, such as Element Distinctness, Set Disjointness, Finding Convex Hull, etc. Through further transformations, it leads to lower bounds for problems such as Integer Max Gap and Closest Pair of a simple polygon. The proof involves a nontrivial extension of the Milnor–Thorn techniques for finding upper bounds on the Betti numbers of algebraic varieties.