On the nullstellensätze for stein spaces and C-analytic sets.



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Acquistapace, Francesca and Broglia, Fabrizio and Fernando Galván, José Francisco (2016) On the nullstellensätze for stein spaces and C-analytic sets. Transactions of the American Mathematical Society, 368 (6). pp. 3899-3929. ISSN 0002-9947

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Official URL: http://www.ams.org/journals/tran/2016-368-06/S0002-9947-2015-06436-8/S0002-9947-2015-06436-8.pdf


In this work we prove the real Nullstellensatz for the ring O(X) of analytic functions on a C-analytic set X ⊂ Rn in terms of the saturation of Łojasiewicz’s radical in O(X): The ideal I(Ƶ(a)) of the zero-set Ƶ(a) of an ideal a of O(X) coincides with the saturation (Formula presented) of Łojasiewicz’s radical (Formula presented). If Ƶ(a) has ‘good properties’ concerning Hilbert’s 17th Problem, then I(Ƶ(a)) = (Formula presented) where (Formula presented) stands for the real radical of a. The same holds if we replace (Formula presented) with the real-analytic radical (Formula presented) of a, which is a natural generalization of the real radical ideal in the C-analytic setting. We revisit the classical results concerning (Hilbert’s) Nullstellensatz in the framework of (complex) Stein spaces. Let a be a saturated ideal of O(Rn) and YRn the germ of the support of the coherent sheaf that extends aORn to a suitable complex open neighborhood of Rn. We study the relationship between a normal primary decomposition of a and the decomposition of YRn as the union of its irreducible components. If a:= p is prime, then I(Ƶ(p)) = p if and only if the (complex) dimension of YRn coincides with the (real) dimension of Ƶ(p).

Item Type:Article
Uncontrolled Keywords:Nullstellensatz; Stein space; Closed ideal; Radical; Real Nullstellensatz; C-analytic set; Saturated ideal; Lojasiewicz’s radical; Convex ideal; H-sets, Ha-set; Real ideal; Real radical; Real-analytic ideal; Real-analytic radical; Quasi-real ideal.
Subjects:Sciences > Mathematics > Functions
ID Code:38181
Deposited On:23 Jun 2016 08:25
Last Modified:23 Jun 2016 08:25

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