Let R be a real closed field, and denote by the ring of germs, at the origin of Rⁿ, of functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring with some natural properties. We prove that, for each n ∈ ℕ, is a noetherian ring and if R = ℝ (the field of real numbers), then , where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap100-3-4,
author = {Abdelhafed Elkhadiri},
title = {On some noetherian rings of $C^{$\infty$}$ germs on a real closed field},
journal = {Annales Polonici Mathematici},
volume = {101},
year = {2011},
pages = {261-275},
zbl = {1226.32005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap100-3-4}
}
Abdelhafed Elkhadiri. On some noetherian rings of $C^{∞}$ germs on a real closed field. Annales Polonici Mathematici, Tome 101 (2011) pp. 261-275. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap100-3-4/