We survey both old and new developments in the theory of algorithms in real algebraic geometry â€“ starting from effective quantifier elimination in the first order theory of reals due to Tarski andâ€¦ (More)

Let R be a real closed field, P,Q âŠ‚ R[X1, . . . , Xk] finite subsets of polynomials, with the degrees of the polynomials in P (resp. Q) bounded by d (resp. d0). Let V âŠ‚ Rk be the real algebraicâ€¦ (More)

In this paper we give a new bound on the sum of the Betti numbers of semi-algebraic sets. This extends a well-known bound due to Oleinik and Petrovsky 19], Thom 23] and Milnor 18]. In separate papersâ€¦ (More)

In this paper, we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now.â€¦ (More)

Let R be a real closed field and let Q and P be finite subsets of R[X1, . . . ,Xk] such that the set P has s elements, the algebraic set Z defined by âˆ§ QâˆˆQ Q = 0 has dimension k â€² and the elements ofâ€¦ (More)

We consider a family ofs polynomials,P = fP ; . . . ; P 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 dimensionk which isâ€¦ (More)

We consider a semi-algebraic set S defined by s polynomials of degree d in k variables. We present a new algorithm for computing a semi-algebraic path in S comecting two points if they happen to lieâ€¦ (More)