On some noetherian rings of $C^{\infty }$ germs on a real closed field

Abdelhafed Elkhadiri. "On some noetherian rings of $C^{∞}$ germs on a real closed field." Annales Polonici Mathematici 100.3 (2011): 261-275. <http://eudml.org/doc/280385>.

@article{AbdelhafedElkhadiri2011,
abstract = {Let R be a real closed field, and denote by $_\{R,n\}$ the ring of germs, at the origin of Rⁿ, of $C^∞$ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring $_\{R,n\} ⊂ _\{R,n\}$ with some natural properties. We prove that, for each n ∈ ℕ, $_\{R,n\}$ is a noetherian ring and if R = ℝ (the field of real numbers), then $_\{ℝ,n\} = ₙ$, 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 $_\{R,n\}$.},
author = {Abdelhafed Elkhadiri},
journal = {Annales Polonici Mathematici},
keywords = {real closed field; Weierstrass division theorem; semi-analytic sets},
language = {eng},
number = {3},
pages = {261-275},
title = {On some noetherian rings of $C^\{∞\}$ germs on a real closed field},
url = {http://eudml.org/doc/280385},
volume = {100},
year = {2011},
}

TY - JOUR
AU - Abdelhafed Elkhadiri
TI - On some noetherian rings of $C^{∞}$ germs on a real closed field
JO - Annales Polonici Mathematici
PY - 2011
VL - 100
IS - 3
SP - 261
EP - 275
AB - Let R be a real closed field, and denote by $_{R,n}$ the ring of germs, at the origin of Rⁿ, of $C^∞$ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring $_{R,n} ⊂ _{R,n}$ with some natural properties. We prove that, for each n ∈ ℕ, $_{R,n}$ is a noetherian ring and if R = ℝ (the field of real numbers), then $_{ℝ,n} = ₙ$, 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 $_{R,n}$.
LA - eng
KW - real closed field; Weierstrass division theorem; semi-analytic sets
UR - http://eudml.org/doc/280385
ER -