An Improved Algorithm for Quantifier Elimination Over Real Closed Fields

In this paper we give a new algorithm for quanti-er elimination in the rst order theory of real closed elds that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms 3, 25, 20] the combinatorial part of the complexity of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depend only on a constant number of the free variables, the algebraic part of the complexity can also be made independent of the number of free variables. This new feature of our algorithm allows us to obtain a new algorithm for a variant of the quantiier elimination problem. We give an almost optimal algorithm for this new problem, which we call the uniform quantiier elimination problem and apply it to solve a problem arising in the eld of constraint databases. No algorithm with reasonable complexity bound was known for this latter problem till now. We also point out interesting logical consequences of this algorithmic result, concerning the expressive power of a constraint query language over the reals. Moreover, our improved algorithm for performing quantiier elimination immediately leads to improved algorithms for several problems for which quantiier elimination is a basic step, for example, the problem of computing the closure of a given semi-algebraic set.