Geometric problems solvable in single exponential time

Abstract

Let S be a semialgebraic set given by a boolean combination of polynomial inequalities. We present an algorithmical method which solves in single exponential sequential time and polynomial parallel time, the following problems: computation of the dimension of S. computation of the number of semialgebraically connected components of S and construction of paths in S connecting points in the same component. computation of the distance of S to another semialgebraic set and finding points realizing the distance if they exist. computation of the “optical resolution” of S if S is closed (the pelotita and the bolon). computation of integer Morse directions of S if S is a regular algebraic hypersurface.