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

• Autorzy: Abdelhafed Elkhadiri
• Języki publikacji: EN

Abstrakty

EN

Let R be a real closed field, and denote by $𝓔_{R,n}$ 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 $𝓐_{R,n} ⊂ 𝓔_{R,n}$ with some natural properties. We prove that, for each n ∈ ℕ, $𝓐_{R,n}$ is a noetherian ring and if R = ℝ (the field of real numbers), then $𝓐_{ℝ,n} = 𝓗ₙ$, where 𝓗ₙ 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 $𝓐_{R,n}$.

Kategorie tematyczne

• 32B20: Semi-analytic sets and subanalytic sets
• 13F25: Formal power series rings
• 03C10: Quantifier elimination, model completeness and related topics
• 26E10: C ∞ -functions, quasi-analytic functions
• 32B05: Analytic algebras and generalizations, preparation theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2011
• Tom: 100
• Numer: 3
• Strony: 261-275

Daty

wydano

2011

Twórcy

autor

Abdelhafed Elkhadiri

• Department of Mathematics, Faculty of Sciences, University Ibn Tofail, B.P. 133, Kénitra, Morocco

Identyfikatory

DOI

10.4064/ap100-3-4