Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1,...,Xk]andSasemi-algebraic subset of Z(Q), defined by a Boolean formula with atoms of the form P < 0,P> 0,P =0withP∈P, where P is a finite subset of R[X1,...,Xk]. A semi-algebraic set C is semi-algebraically connected if it is non-empty and is not

We consider a family of s polynomials, P = fP 1 ; : : : ; P s g; in k variables with coefficients in a real closed field R, each of degree at most d, and an algebraic variety V of real dimension k 0 which is defined as the zero set of a polynomial Q of degree at most d. The number of semi-algebraically connected components of all nonempty sign condition on P over V is bounded by \Gamma O(s) k 0 \Delta d O(k) (see [4]). In this paper we present a new algorithm to compute a set of points meeting every semi-algebraically connected component of each non empty sign condition of P over V . Its complexity is \Gamma O(s) k 0 \Delta sd O(k) . This interpolates a sequence of results between the algorithm of BenOr, Kozen and Reif [5] which is the case k 0 = 0, in one variable, and the algorithm of Basu-Pollack-Roy [1] which is the case k 0 = k. It Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A.. Supported in part by NSF grants CCR...

In this thesis we present new algorithms to solve several very general problems of semi-algebraic geometry. These are currently the best algorithms for solving these problems. In addition, we have proved new bounds on the topological complexity of real semi-algebraic sets, in terms of the parameters of the polynomial system defining them, which improve some old and widely used results in this field. In the first part of the thesis we describe new algorithms for solving the decision problem for the first order theory of real closed fields and the more general problem of quantifier elimination. Moreover, we prove some purely mathematical theorems on the number of connected components and on the existence of small rational points in a given semi-algebraic set. The second part of this thesis deals with connectivity questions of semialgebraic sets. We develop new techniques in order to give a...

We consider a family of s polynomials, P = fP 1 ; : : : ; P s g; in k variables with coefficients in a real closed field R, each of degree at most d, and an algebraic variety V of real dimension k 0 which is defined as the zero set of a polynomial Q of degree at most d. The number of semi-algebraically connected components of all nonempty sign conditions on P over V is bounded by s k 0 (O(d)) k (see [4]). In this paper we present a new algorithm to compute a set of points meeting every semi-algebraically connected component of each non empty sign condition of P over V . Its complexity is s k 0 +1 d O(k) . This interpolates a sequence of results between the algorithm of BenOr, Kozen and Reif [5] which is the case k 0 = 0, in one variable, and the algorithm of Basu-Pollack-Roy [1] which is the case k 0 = k. It improves the results in [2] where the same problem was solved in time s k 0 +1 d O(k 0 k) . Courant Institute of Mathematical Sciences, New York Univ...