References
Artin, E.: Über die Zerlegung definiter Funktionen in Quadrate. Hamb. Abh.5, 100–115 (1927); Collected Papers, 273–288, Reading MA: Addison-Wesley 1965
Artin, M., Mazur, B.: On a periodic points. Ann. of Math.81, 82–99 (1965)
Bourbaki, N.: Algébre, groupes et corps ordonnes, Chap. 6, Sect. 2. Paris: Hermann 1952
Bochnak, J.: Sur le 17ême probleme de Hilbert pour les fonctions de Nash. Proc. Am. Math. Soc.71, 183–189 (1978)
Bochnak, J.: Sur la factorialité des anneaux de fonctions de Nash. Comment. Math. Helv.52, 211–218 (1977)
Bochnak, J.: Sur la factorialité de fonctions analytiques. C. R. Acad. Sci. Paris279, 269–273 (1974)
Bochnak, J., Risler, J.J.: Sur le théorème de zéros pour les variétés analytiques réeles de dimension 2. Ann. Sci. École Norm. Sup.8, 353–364 (1975)
Bochnak, J., Risler, J.J.: Analyse différentielle et géometrie analytique. In: Lecture Notes in Mathematics, Vol. 535, pp. 63–69. Berlin, Heidelberg, New York: Springer 1976
Choi, M.D., Lam, T.Y.: Extremal semidefinite forms. Math. Ann.231, 1–18 (1977)
Cohen, P.: Decision procedure for real andp-adic fields. Comm. Pure Appl. Math.22, 131–151 (1969)
Dieudonné, J.: Cours de géometrie algebrique. Vols. I and II. Paris: Hermann 1974
Dieudonné, J., Carrell, J.: Invariant theory old and new. New York: Academic Press 1971
Dubois, D., Efroymson, G.: Algebraic theory of real varieties. I. Studies and Essays presented to Y.-H. Chen. Taiwan Math. J. 107–135 (1970)
Efroymson, G.: A Nullstellensatz for Nash rings. Pacific J. Math.54, 101–112 (1974)
Efroymson, G.: Substitution in Nash functions. Pacific J. Math.63, 137–145 (1976)
Galbiati, M., Tognoli, A.: Alcuna proprieta delle varieta algebriche reali. Ann. Scuola Norm. Sup. Pisa Cl. Sci.27, 359–404 (1972)
Gondard, D., Ribenboim, P.: Fonctions definies positives sur les varietes reeles. Bull. Sci. Math.98, 39–47 (1974)
Grothendieck, A., Dieudonné, J.: Élements de géometrie algébrique: Étude Locale des Schémas et des Morphismes de Schémas. I.H.E.S. Publ. Math.24, 1965
Hironaka, H.: Triangulation of algebraic sets. Proc. of Symp. Pure Math.29, 165–186 (1975)
Hironaka, H.: Subanalytic sets, Vol. in honor of Y. Akizuki. In: Number theory, algebraic geometry, and commutative algebra, pp. 453–493. Tokyo: Kinokuniya 1973
Knebusch, M.: On algebraic curves over real closed fields. I, II. Math. Z.150, 49–70 (1976);151, 189–205 (1976)
Lazzeri, F., Tognoli, A.: Alcune proprieta degli spazi algebrici. Ann. Scuola Norm. Sup. Pisa Cl. Sci.24, 597–632 (1970)
Lönsted, K.: An algebraization of vector bundles on compact manifolds. J. Pure Appl. Algebra2, 193–207 (1972)
Luna, D.: Fonctions differentiables invariants sous l'action d'un groupe reductif. Ann. Inst. Fourier (Grenoble)26, 33–49 (1976)
Mostowski, T.: Some properties of the ring of Nash functions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. III, 243–266 (1976)
Motzkin, T.S.: The arithmetic-geometric inequality. In: Inequalities, pp. 205–224. Shisha, O. ed. New York: Academic Press 1967
Mumford, D.: Introduction to algebraic geometry, Preliminary version of first three chapters. London: Harvard University Press 1969
Nash, J.: Real algebraic manifolds. Ann. of Math.56, 4, 405–421 (1952)
Palais, R.: Equivariant real algebraic differential topology, Part I. Smoothness categories and Nash manifolds. Notes, Brandeis University 1972
Pfister, A.: Hilbert's 17th Problem and related problems. Proc. Symp. Pure Math.28, pp. 483–489. Providence: American Mathematical Society 1976
Procesi, C.: Positive symmetric functions. Advances in Math.29, 219–225 (1978)
Risler, J.J.: Sur l'anneau des fonctions de Nash globales. Ann. Sci. Ecole Norm. Sup.8, 365–378 (1975)
Risler, J.J.: Les théorèmes des zéros en géometrie algébriques et analytique réeles. Bull. Soc. Math. France104, 113–127 (1976)
Robinson, A.: Introduction to model theory and to the mathematics of algebra. Amsterdam: North Holland 1963
Shiota, M.: On the unique factorization of the ring of Nash functions. Publ. R.I.M.S. Kyoto Univ. (to appear)
Stengle, G.: Nullstellensatz and Positivestellensatz in semi-algebraic geometry. Math. Ann.207, 87–97 (1974)
Swan, R.: Topological examples of projective modules. Trans. A.M.S.230, 201–234 (1977)
Thom, R.: Stabilité structuelle et morphogenese. Benjamin: New York: 1972
Tougeron, J.C.: Idéaux des fonctions differentiables. In: Ergebnisse der Mathematik, Vol. 71. Berlin, Heidelberg, New York: Springer 1972
Bochnak, J., Kucharz, W., Shiota, M.: On equivalence of ideals of real global analytic functions and the 17th Hilbert problem (to appear)
Bochnak, J., Shiota, M.: The divisor class group of a ring of real global analytic, Nash or regular rational functions (to appear)
Brumfiel, G.W.: Partially ordered rings and semi-algebraic geometry. Cambridge: Cambridge University Press 1979
Dubois, A.: A Nullstellensatz for ordered fields. Ark. Mat8, 111–114 (1969)
Efroymson, G.: Nash rings of planar domains. Trans. AMS249(2), 435–445 (1979)
Kaplansky, J.: Hilbert's Problems. Preliminary Edition, University of Chicago, 163–181, 1977
Lojasiewicz, S.: Ensembles semi-analytiques. I.H.E.S., Bures-sur-Yvette, preprint 1965
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bochnak, J., Efroymson, G. Real algebraic geometry and the 17th Hilbert problem. Math. Ann. 251, 213–241 (1980). https://doi.org/10.1007/BF01428942
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01428942