Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2010, November 26.

Abstract:
Given a quasianalytic system ${\cal Q} =({\cal Q}_{n})_{n\in \matN}$ of sheaves, denote by $Q_{n}$ the local ring of Q-analytic function germs at $0 \in \matR^{n}$. This paper introduces the concepts of \L{}ojasiewicz radical and geometric spectrum $\mbox{Speg}\, Q_{n} \subset \mbox{Sper}\, Q_{n}$. Via the \L{}ojasiewicz inequality, a version of the Nullstellensatz for $Q_{n}$ is given. We establish a quasianalytic version of the Artin--Lang property for $Q_{n}$. Finally, we prove, by means of transformation to normal crossings by blowing up, that the \L{}ojasiewicz radical $\pounds(I)$ of any ideal $I \subset Q_{n}$ coincides with the contraction of the real radical $\Re(I\widehat{Q_{n}})$.

Mathematics Subject Classification (2010): 26E10, 14P15, 13J30.

Keywords and Phrases: quasianalytic functions, semi-quasianalytic sets, Artin--Lang property, Lojasiewicz radical, real Nullstellensatz.

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