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On a property of ideals of differentiable functions

Bruno, Oscar P. (1986) On a property of ideals of differentiable functions. Bulletin of the Australian Mathematical Society, 33 (2). pp. 293-305. ISSN 0004-9727. https://resolver.caltech.edu/CaltechAUTHORS:20181106-160538496

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Abstract

Let J ⊆ C^∞(R^n) be any ideal. Since a function of the variables t = (t_1,...,t_n) is a function of the variables (t,x) = (t_1,...,t_n,x_1,...x_p) which does not depend on x, we have J ⊆ C^∞(R^(n+p)). Of course, J is not an ideal of C^∞(R^(n+p), but it generates an ideal that we call J(t,x). Consider the following statement (1) on J: “Given any f ϵ C^∞ (R^(n+p), f ϵ J(t,x) if and only if for every fixed a ϵ R^p, f(t,a) ϵ J". In this paper we show that statement (1) holds for a large class of finitely generated ideals although not for all of them. We say that ideals satisfying statement (1) have line determined extensions. We characterize these ideals to be closed ideals J(t) (in the sense of Whitney) such that for all p ∈ ℕ, the ideal J(t,x) is also closed. Finally, some non-trivial examples are developed.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1017/S0004972700003166DOIArticle
ORCID:
AuthorORCID
Bruno, Oscar P.0000-0001-8369-3014
Additional Information:© 1986 Australian Mathematical Society. Received 16 July 1985.
Issue or Number:2
Record Number:CaltechAUTHORS:20181106-160538496
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20181106-160538496
Official Citation:Bruno, O. (1986). On a property of ideals of differentiable functions. Bulletin of the Australian Mathematical Society, 33(2), 293-305. doi:10.1017/S0004972700003166
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:90685
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:07 Nov 2018 19:32
Last Modified:03 Oct 2019 20:27

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