(Joint work with N. Vorobjov)
Upper bounds for the Betti numbers of real algebraic sets were obtained by Oleinik-Petrovskii (1949), Milnor (1964) and Thom (1965). These bounds, based on Morse theory, were single exponential in the number of variables. Basu (1999) extended these results to real semialgebraic sets defined by equations and non-strict inequalities. However, the best previously known upper bounds for general semialgebraic sets were double exponential. Gabrielov and Vorobjov (2005) obtained a single exponential upper bound on the Betti numbers of a general semialgebraic set. Given a semialgebraic set X, another semialgebraic set Y defined by equations and non-strict inequalities is constructed, with the same Betti numbers as X. Basu's theorem applied to Y provides the upper bound for the Betti numbers of X. The method easily generalizes to non-algebraic functions, such as Pfaffian functions.