Titlebook: Real Algebraic Geometry; Jacek Bochnak,Michel Coste,Marie-Françoise Roy Book 1998 Springer-Verlag Berlin Heidelberg 1998 Nash functions.Na

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The Tarski-Seidenberg Principle as a Transfer Tool,ing .. We then explore further the Tarski-Seidenberg principle. It has already been used in two different ways: to prove that the projection of a semi-algebraic set is semi-algebraic (in Chap. 2), and to obtain the Artin-Lang homomorphism theorem (in Chap. 4). Here we study some additional applicati

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,Hilbert’s 17th Problem. Quadratic Forms in Real Algebraic Geometry,mial over a real closed field is a sum of squares of rational functions. We also prove a few generalizations. The second section deals with the equivariant version of Hilbert’s 17. problem. In the third section, we address the problem of representing nonnegative polynomials as sum of squares of poly

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Real Spectrum,us defined is the real spectrum of the ring. In the first section, we give some general properties of the real spectrum of a ring. Some of these properties are analogous to the properties of the Zariski spectrum (compactness, for instance), while others are more specific (for instance, the fact that

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Nash Functions,ebraic over the polynomials, and (for . = ℝ) the ring of germs of analytic algebraic functions. Prom the study of the ring of algebraic power series in the second section, we deduce the local properties of Nash functions. In the third section, we state a theorem of approximation of formal solutions

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Stratifications,l structure with respect to all successive projections .. → .., which is particularly useful in inductive arguments. In the second section, we prove that a closed and bounded semi-algebraic set can be semi-algebraically triangulated. In Section 3, we investigate the structure of continuous semi-alge

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Topology of Real Algebraic Varieties,s the fact that, for every semi-algebraic triangulation of a bounded algebraic set of dimension . and every (. - 1)-simplex . of such a triangulation, the number of .-simplices of the triangulation having . as a face is even. In the second section, we use this property and an appropriate stratificat

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Polynomial or Regular Mappings with Values in Spheres, The results concerning this comparison are very diverse. Section 1 deals with the existence of nonconstant polynomial mappings from .. into .., using mainly the theory of quadratic forms. We prove Wood’s theorem, which states that if . is a power of 2 and . > n, then every polynomial mapping .. → .

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Witt Rings in Real Algebraic Geometry,ver a real closed field ., we show that the Witt ring .(..(.)) coincides with ..(..(.)) (≃ .(.) if . = ℝ). The second section is devoted to the result of Mahé concerning the separation of the semi-algebraically connected components of . by the signatures of elements of .(.(.)). In the third section