1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem.… Expand

This chapter discusses computing roadmaps and Connected Components of Algebraic Sets, as well as the "complexity of Basic Algorithms" and "cylindrical Decomposition Algorithm".Expand

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 is given.Expand

A new exact divisibility for sub-resultants in the defective case is presented which extends the formulae for the non-defective situation in a natural way and the size of coefficients in the ordinary remainder sequence is quadratic in d.Expand

We want to propose alternative computational methods for dealing with the following three classical problems in the study of zero-dimensional systems, rephrased in the context of finite-dimensional… Expand

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 is given.Expand

An algorithm is proposed that finds, when the algebraic set is non empty, at least one point on each semi-algebraically connected component of the set when the set is not empty.Expand

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] and S a semi-algebraic subset of Z(Q), defined by a Boolean formula with… Expand