Analytic and differentiable functions vanishing on an algebraic set
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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
- William A. Adkins and J. V. Leahy, Criteria for finite generation of ideals of differentiable functions, Duke Math. J. 42 (1975), no. 4, 707–716. MR 400287
- William A. Adkins and J. V. Leahy, A global real analytic nullstellensatz, Duke Math. J. 43 (1976), no. 1, 81–86. MR 396991
- Joseph Becker, Parametrizations of analytic varieties, Trans. Amer. Math. Soc. 183 (1973), 265–292. MR 344513, DOI 10.1090/S0002-9947-1973-0344513-8
- Jacek Bochnak and Jean-Jacques Risler, Analyse différentielle et géométrie analytique. Quelques questions ouvertes, Singularités d’applications différentiables (Sém., Plans-sur-Bex, 1975) Lecture Notes in Math., Vol. 535, Springer, Berlin, 1976, pp. 63–69 (French). MR 0464288
- Jacek Bochnak and Jean-Jacques Risler, Sur la divisibilité des fonctions différentiables, Singularités d’applications différentiables (Sém., Plans-sur-Bex, 1975) Lecture Notes in Math., Vol. 535, Springer, Berlin, 1976, pp. 45–62 (French). MR 0477117
- Otto Forster, Zur Theorie der Steinschen Algebren und Moduln, Math. Z. 97 (1967), 376–405 (German). MR 213611, DOI 10.1007/BF01112815
- B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
- Jean Merrien, Faisceaux analytiques semi-cohérents, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 4, 165–219 (French). MR 599629
- Walter Rudin, A geometric criterion for algebraic varieties, J. Math. Mech. 17 (1967/1968), 671–683. MR 0219750
- A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2) 60 (1954), 365–374. MR 63994, DOI 10.2307/1969640
- Yum-tong Siu, Noetherianness of rings of holomorphic functions on Stein compact series, Proc. Amer. Math. Soc. 21 (1969), 483–489. MR 247135, DOI 10.1090/S0002-9939-1969-0247135-7
- Jean-Claude Tougeron, Idéaux de fonctions différentiables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71, Springer-Verlag, Berlin-New York, 1972. MR 0440598
Additional Information
- © Copyright 1988 American Mathematical Society
- 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