On the Nullstellensätze for Stein spaces and $C$-analytic sets

Abstract:

In this work we prove 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 saturation of Łojasiewicz’s radical in $\mathcal {O}(X)$: 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 $\widetilde {\sqrt [L]{\mathfrak {a}}}$ of Łojasiewicz’s radical $\sqrt [L]{\mathfrak {a}}$. 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 real radical of $\mathfrak {a}$. The same holds if we replace $\sqrt [\mathsf {r}]{\mathfrak {a}}$ with the real-analytic radical $\sqrt [\mathsf {ra}]{\mathfrak {a}}$ of $\mathfrak {a}$, which is a natural generalization of the real radical ideal in the $C$-analytic setting. We revisit the classical results concerning (Hilbert’s) Nullstellensatz in the framework of (complex) Stein spaces.

Let $\mathfrak {a}$ be a saturated ideal of $\mathcal {O}(\mathbb {R}^n)$ and $Y_{\mathbb {R}^n}$ the germ of the support of the coherent sheaf that extends $\mathfrak {a}\mathcal {O}_{\mathbb {R}^n}$ to a suitable complex open neighborhood of $\mathbb {R}^n$. We study the relationship between a normal primary decomposition of $\mathfrak {a}$ and the decomposition of $Y_{\mathbb {R}^n}$ as the union of its irreducible components. If $\mathfrak {a}:=\mathfrak {p}$ is prime, then $\mathcal {I}(\mathcal {Z}(\mathfrak {p}))=\mathfrak {p}$ if and only if the (complex) dimension of $Y_{\mathbb {R}^n}$ coincides with the (real) dimension of $\mathcal {Z}(\mathfrak {p})$.