Lower bounds for algebraic computation trees with integer inputs

A proof is given of a general theorem showing that for certain sets W a certain topological lower bound is valid in the algebraic computation tree model, even if the inputs are restricted to be integers. The theorem can be used to prove tight lower bounds for the integral-constrained form of many basic problems, such as element distinctness, set disjointness, and finding the convex hull. Through further transformations it leads to lower bounds for problems such as the integer max gap and closest pair of a simple polygon. The proof involves a nontrivial extension of the Milnor-Thom techniques for finding upper bounds on the Betti numbers of algebraic varieties.