Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Analytic and differentiable functions vanishing on an algebraic set


Author: Wojciech Kucharz
Journal: Proc. Amer. Math. Soc. 102 (1988), 514-516
MSC: Primary 32B15; Secondary 26B99, 32B05
DOI: https://doi.org/10.1090/S0002-9939-1988-0928970-8
MathSciNet review: 928970
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ U$ be an open semi-algebraic subset of $ {{\mathbf{R}}^n}$ and let $ X$ be a closed analytic subset of $ U$ which is also a semi-algebraic set (e.g., $ U = {{\mathbf{R}}^n}$ and $ X$ is an algebraic subset of $ {{\mathbf{R}}^n}$). It is proved that the ideal of analytic functions on $ U$ vanishing on $ X$ is finitely generated provided that the set $ X$ is coherent. The ideal of infinitely differentiable functions on $ U$ vanishing on $ X$ is finitely generated if and only if the set $ X$ is coherent.


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

  • [1] W. A. Adkins and J. V. Leahy, Criteria for finite generation of ideals of differentiable functions, Duke Math. J. 42 (1975), 707-716. MR 0400287 (53:4122)
  • [2] -, A global real analytic nullstellensatz, Duke Math. J. 43 (1976), 81-86. MR 0396991 (53:851)
  • [3] J. Becker, Parametrizations of analytic varieties, Trans. Amer. Math. Soc. 183 (1973), 265-292. MR 0344513 (49:9252)
  • [4] J. Bochnak and J. J. Risler, Analyse différentiale et géométrie analytique, quelques questions ouvertes, singularités d'applications différentiables, Lecture Notes in Math., vol. 535, Springer, 1976, pp. 63-69. MR 0464288 (57:4222)
  • [5] -, Sur la divisibilité des fonctions différentiables, singularités d'applications différentiables Lecture Notes in Math., vol. 535, Springer, 1976, pp. 45-62. MR 0477117 (57:16661)
  • [6] O. Forster, Zur Theorie der Steinichen Algebren und Moduln, Math. Z. 97 (1967), 376-405. MR 0213611 (35:4469)
  • [7] B. Malgrange, Ideals of differentiable functions, Oxford Univ. Press, 1966. MR 0212575 (35:3446)
  • [8] J. Merrien, Faisceaux analytiques semi-cohérents, Ann. Inst. Fourier (Grenoble) 30 (1980), 165-219. MR 599629 (82f:32018)
  • [9] W. Rudin, A geometric criterion for algebraic varieties, J. Math. Mech. 17 (1968), 671-683. MR 0219750 (36:2829)
  • [10] A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2) 60 (1954), 365-374. MR 0063994 (16:209a)
  • [11] Y. T. Siu, Noetherianness of rings of holomorphic functions on Stein compact subsets, Proc. Amer. Math. Soc. 21 (1969), 483-489. MR 0247135 (40:404)
  • [12] J. Cl. Tougeron, Idéaux de fonctions différentiables, Ergebnisse der Math. 71, Springer-Verlag, 1972. MR 0440598 (55:13472)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32B15, 26B99, 32B05

Retrieve articles in all journals with MSC: 32B15, 26B99, 32B05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0928970-8
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society