On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals

This series of papers presents a complete development and complexity analysis of a decision method, and a quantifier elimination method, for the first order theory of the reals. The complexity upper bounds which are established are the best presently available, both for sequential and parallel computation, and both for the bit model of computation and the real number model of computation; except for the bounds pertaining to the sequential decision method in the bit model of computation, all bounds represent significant improvements over previously established bounds.