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Sur le 17ème problème de Hilbert pour les fonctions de Nash


Author: Jacek Bochnak
Journal: Proc. Amer. Math. Soc. 71 (1978), 183-188
MSC: Primary 32B05; Secondary 12D99
DOI: https://doi.org/10.1090/S0002-9939-1978-0486597-2
MathSciNet review: 0486597
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Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this note is to give a more refined version of a theorem of Efroymson: If $ U \subset {{\mathbf{R}}^n}$ is defined by polynomial inequalities of the form $ {f_i} > 0,i = 1, \ldots ,p$, and if g is a positive definite Nash function on U, then g is a finite sum of squares of Nash meromorphe functions on U.


References [Enhancements On Off] (What's this?)

  • [1] Emil Artin, The collected papers of Emil Artin, Edited by Serge Lang and John T. Tate, Addison–Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. MR 0176888
  • [2] Jacek Bochnak, Sur la factorialité des anneaux de fonctions de Nash, Comment. Math. Helv. 52 (1977), no. 2, 211–218 (French). MR 0457763, https://doi.org/10.1007/BF02567365
  • [3] J. Bochnak and J.-J. Risler, Le théorème des zéros pour les variétés analytiques réelles de dimension 2, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 3, 353–363. MR 0396994
  • [4] Paul J. Cohen, Decision procedures for real and 𝑝-adic fields, Comm. Pure Appl. Math. 22 (1969), 131–151. MR 0244025, https://doi.org/10.1002/cpa.3160220202
  • [5] Gustave A. Efroymson, A Nullstellensatz for Nash rings, Pacific J. Math. 54 (1974), 101–112. MR 0360576
  • [6] Gustave Efroymson, Substitution in Nash functions, Pacific J. Math. 63 (1976), no. 1, 137–145. MR 0409456
  • [7] D. Gondard, Le 17ème problème de Hilbert, Thèse du 3ème cycle, Paris Orsay 1974.
  • [8] S. Land, Algèbre, Addison-Wesley, Reading, Mass., 1965.
  • [9] Tadeusz Mostowski, Some properties of the ring of Nash functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 2, 245–266. MR 0412180
  • [10] Albrecht Pfister, Hilbert’s seventeenth problem and related problems on definite forms, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Un1974), Amer. Math. Soc., Providence, R.I., 1976, pp. 483–489. MR 0424679
  • [11] J.-J. Risler, Les théorèmes des zeros en géométries algébrique et analytique réelles, Bull. Soc. Math. France 104 (1976), 113-127.
  • [12] N. Bourbaki, Algèbre, Chapitre 6, Groupes et corps ordonnés, $ \S2$, Hermann, Paris, 1952.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0486597-2
Keywords: 17th Hilbert problem, Nash functions, Tarski principle, semi-algebraic sets, real closed field
Article copyright: © Copyright 1978 American Mathematical Society

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