20 sep 2016 -- 14:30
Aula Tricerri, DiMaI "Dini", Firenze
Abstract.
(joint work with: Francesca Acquistapace, Fabrizio Broglia)
In this seminar we present the real Nullstellensatz for the ring ${\mathcal O}(X)$ of analytic functions on a $C$-analytic set $X\subset{\mathbb R}^n$ in terms of the \em saturation \em of \L ojasiewicz's radical in ${\mathcal O}(X)$: \em The ideal ${\mathcal I}({\mathcal Z}({\mathfrak a}))$ of the zero-set ${\mathcal Z}({\mathfrak a})$ of an ideal ${\mathfrak a}$ of ${\mathcal O}(X)$ coincides with the saturation $\widetilde{\sqrt[\text{\L}]{{\mathfrak a}}}$ of \L ojasiewicz's radical $\sqrt[\text{\L}]{{\mathfrak a}}$\em. If ${\mathcal Z}({\mathfrak a})$ has `good properties' concerning Hilbert's 17th Problem, then ${\mathcal I}({\mathcal Z}({\mathfrak a}))=\widetilde{\sqrt[\mathsf{r}]{{\mathfrak a}}}$ where $\sqrt[\mathsf{r}]{{\mathfrak a}}$ stands for the \em real radical \em of ${\mathfrak a}$. The same holds if we replace $\sqrt[\mathsf{r}]{{\mathfrak a}}$ with the \em real-analytic radical \em $\sqrt[\mathsf{ra}]{{\mathfrak a}}$ of ${\mathfrak a}$, which is a natural generalization of the real radical ideal in the $C$-analytic setting. We also revisit the classical results concerning (Hilbert's) Nullstellensatz in the framework of (complex) Stein spaces.
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