Titlebook: Algorithms in Real Algebraic Geometry; Saugata Basu,Richard Pollack,Marie-Franco̧ise Roy Textbook 20031st edition Springer-Verlag Berlin H

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Real Closed Fields,described in the second section. The fourth section is devoted to several important applications of the projection theorem, of logical and geometric nature. In the last section, an important example of a non-archimedean real closed field is described: the field of Puiseux series.

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Semi-Algebraic Sets,itute a real closed field containing infinitesimals, closely related to the field of Puiseux series, and play an important role throughout the whole book. We end the chapter with a section on semi-algebraic differentiable functions.

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Elements of Topology,r homology that applies only to simplicial complexes. In the second section, we show how to extend this theory to closed semi-algebraic sets using the triangulation theorem proved in Chapter 5. Finally, in the third section we define the Euler-Poincare characteristic for locally closed semi-algebraic sets.

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Complexity of Basic Algorithms,atic forms. In Section 3, we study remainder sequences and the related notion of subresultant polynomials. The algorithms in this chapter are very basic and will be used throughout the other chapters of the book.

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Real Roots,y for archimedean real closed fields. In the second part of the chapter we study exact methods working in general real closed fields. Section 3 is devoted to exact sign determination in a real closed field and Section 4 to characterizations of roots in a real closed field.