On the combinatorial and algebraic complexity of quantifier elimination

In this paper we give a new algorithm for performing quantifier elimination from first order formulae over real closed fields. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of our algorithm is that the role of the algebraic part (the dependence on the degrees of the input polynomials) and the combinatorial part (the dependence on the number of polynomials) are separated, making possible our improved complexity bound. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that we output, are independent of the number of input polynomials. As special cases of this algorithm, we obtain new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields. Using the theory developed in this paper, we also give an improved bound on the radius of a ball centered at the origin, which is guaranteed to intersect every connected component of the sign partition induced by a family of polynomials. We also use our methods to obtain algorithms for solving certain decision problems in real and complex geometry which improves the complexity of the currently known algorithms for these problems