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Proceedings of the American Mathematical Society
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
MathSciNet review: 928970
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0928970-8
PII: S 0002-9939(1988)0928970-8
Article copyright: © Copyright 1988 American Mathematical Society