Abstract
In the first section, we define real and real closed fields and state some of their basic properties. In the third section, we define semi-algebraic sets and prove that the projection of a semi-algebraic set is semi-algebraic. This is done using a parametric version of real root counting techniques 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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Bibliographical Notes
E. Artin, O. Schreier, Algebraische Konstruktion reeler Körper, Hamb. Abh. 5 8(–99 (1925). The collected papers of Emil Artin, 258–271. Addison-Wesley (1965).
E. Artin, Ober die Zerlegung definiter Funktionen in Quadrate, Hamb. Abh. 5, 100–115 (1927). The collected papers of Emil Artin, 273–288. Reading: Addison-Wesley (1965).
R. Descartes, Géométrie (1636). A source book in Mathematics, 90–31. Harvard University press (1969).
F. Budan de Boislaurent, Nouvelle méthode pour la résolution des équations numériques d’un degré quelconque, (1807), 2nd edition, Paris (1822).
J. Fourier, Analyse des équations déterminées, F. Didot, Paris (1831).
C. Sturm, Mémoire sur la résolution des équations numériques. Inst. France Sc.Math. Phys.6 (1835).
J. J. Sylvester, On a theory of syzygetic relations of two rational integral functions,comprising an application to the theory of Sturm’s function. Trans. Roy. Soc. London (1853).
A. Cauchy, Calcul des indices des fonctions, Journal de l’Ecole Polytechnique, vol. 15, cahier 25, 176–229 (1832).
G. E. Collins, J. Johnson, Quantifier elimination and the sign variation method for real root isolation, ACM SYGSAM Symposium on Symbolic and Algebraic Computations 264–271 (1989).
A. Tarski, A Decision method for elementary algebra and geometry, University of California Press (1951).
A. Tarski, Sur les ensembles définissables de nombres réels, Fund. Math. 17, 210–239 (1931).
A. Seidenberg, A new decision method for elementary algebra, Annals of Mathematics, 60:365–374, (1954).
P. J. Cohen, Decision procedures for real and p-adic fields, Comm. Pure. Appl.Math. 22, 131–151 (1969).
L. Hormander, The analysis of linear partial differential operators, vol. 2. Berlin etc.: Springer-Verlag (1983).
I. Newton, The mathematical papers of Isaac Newton, Cambridge University Press (1968,1971,1976).
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Basu, S., Pollack, R., Roy, MF. (2003). Real Closed Fields. In: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05355-3_3
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DOI: https://doi.org/10.1007/978-3-662-05355-3_3
Publisher Name: Springer, Berlin, Heidelberg
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