Improved Algorithms for Sign Determination and Existential Quantifier Elimination

Recently there has been a lot of activity in algorithms that work over real closed elds, and that perform such calculations as quanti er elimination or computing connected components of semi-algebraic sets. A cornerstone of this work is a symbolic sign determination algorithm due to Ben-Or, Kozen and Reif [2]. In this paper we describe a new sign determination method based on the earlier algorithm, but with two advantages: (i) It is faster in the univariate case, and (ii) In the general case, it allows purely symbolic quanti er elimination in pseudo-polynomial time. By purely symbolic, we mean that it is possible to eliminate a quanti ed variable from a system of polynomials no matter what the coe cient values are. The previous methods required the coe cients to be themselves polynomials in other variables. Our new method allows transcendental functions or derivatives to appear in the coe cients. Another corollary of the new sign-determination algorithm is a very simple, practical algorithm for deciding existentially-quanti ed formulae of the theory of the reals. We present an algorithm that has a bit complexity of nd(c log n) ) randomized, or nd ) c (1+ ) deterministic, for any > 0. Where n is the number of polynomial constraints in the de ning formula, k is the number of variables, d is a bound on the degree, c bounds the bit length of the coe cients. The algorithm makes no general position assumptions, and its constants are much smaller than other recent quanti er elimination methods.