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

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Real Spectrum,ebraic properties of the real spectrum of the ring of polynomial functions on .. In Section 3, we define the value of a semi-algebraic function at a point of the real spectrum; we also show that the continuous semi-algebraic functions are the sections of a sheaf on the real spectrum. Section 4 deals

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Polynomial or Regular Mappings with Values in Spheres,rnionic projective lines. The theory of algebraic vector bundles developed in the previous chapter plays a crucial role. For example, from the fact that every topological (?, ? or ?) line bundle over .. is isomorphic to an algebraic one, we deduce that ?(....) is dense in ....... for . = 1,2,4. In S

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Introduction,ry is the study of real algebraic sets i.e. subsets of .. defined by polynomial equations. By means of a simple example one can see some features which point up the difference between real and complex algebraic geometry. Let us consider the intersection of the straight line . = ., depending on the p

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Semi-algebraic Sets,d inequalities. This class of sets has a remarkable property: stability under projection. Several applications of this basic property are investigated. The study of semi-algebraic sets is based mainly on the ?slicing? technique, which makes it possible to decompose them into a finite number of subse

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Real Algebraic Varieties,ic varieties. In fact, we shall be concerned almost exclusively with affine real algebraic varieties, i.e. real algebraic sets ?up to a biregular isomorphism∝ The third section concerns the notion of nonsingularity. In addition to recalling some properties of varieties valid over an arbitrary field

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Real Algebra,tellensatz, which characterizes the ideals of polynomials vanishing on an algebraic set, when the ground field is real closed. The next two sections develop an ?Artin-Schreier theory? for rings, which is an extension of the theory presented in Chap. 1 for fields. The notion of a prime cone of a ring