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In this work we study some algebraic and topological properties of the ring ${\mathcal O}(X^\nu)$ of global analytic functions of the normalization $(X^\nu,{\mathcal O}_{X^\nu})$ of a reduced complex analytic space $(X,{\mathcal O}_X)$. If $(X,{\mathcal O}_X)$ is a Stein space, we characterize ${\mathcal O}(X^\nu)$ in terms of the (topological) completion of the integral closure $\overline{{\mathcal O}(X)}^\nu$ of the ring ${\mathcal O}(X)$ of global holomorphic functions on $X$ (inside its total ring of fractions) with respect to the usual Fr\'echet topology of $\overline{{\mathcal O}(X)}^\nu$. This shows that not only the Stein space $(X,{\mathcal O}_X)$ but also its normalization is completely determined by the ring ${\mathcal O}(X)$ of global analytic functions on $X$. This result was already proved in 1988 by Hayes-Pourcin when $(X,{\mathcal O}_X)$ is an irreducible Stein space whereas in this paper we afford the general case. We also analyze the real underlying structures $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ and $(X^{\nu\,{\mathbb R}},{\mathcal O}_{X^\nu}^{\mathbb R})$ of a reduced complex analytic space $(X,{\mathcal O}_X)$ and its normalization $(X^\nu,{\mathcal O}_{X^\nu})$. We prove that the complexification of $(X^{\nu\,{\mathbb R}},{\mathcal O}_{X^\nu}^{\mathbb R})$ provides the normalization of the complexification of $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ if and only if $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ is a coherent real analytic space. Roughly speaking, coherence of the real underlying structure is equivalent to the equality of the following two combined operations: (1) normalization + real underlying structure + complexification, and (2) real underlying structure + complexification + normalization.

In this work we present the concept of $C$-semianalytic subset of a real analytic manifold and more generally of a real analytic space. $C$-semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by Cartan ($C$-analytic sets for short). More precisely $S$ is a $C$-semianalytic subset of a real analytic space $(X,{\mathcal O}_X)$ if each point of $X$ has a neighborhood $U$ such that $S\cap U$ is a finite boolean combinations of global analytic equalities and strict inequalities on $X$. By means of paracompactness $C$-semianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on $X$. The family of $C$-semianalytic sets is closed under the same operations as the family of semianalytic sets: locally finite unions and intersections, complement, closure, interior, connected components, inverse images under analytic maps, sets of points of dimension $k$, etc. although they are defined involving only global analytic functions. In addition, we characterize subanalytic sets as the images under proper analytic maps of $C$-semianalytic sets. We prove also that the image of a $C$-semianalytic set $S$ under a proper holomorphic map between Stein spaces is again a $C$-semianalytic set. The previous result allows us to understand better the structure of the set $N(X)$ of points of non-coherence of a $C$-analytic subset $X$ of a real analytic manifold $M$. We provide a global geometric-topological description of $N(X)$ inspired by the corresponding local one for analytic sets due to Tancredi-Tognoli (1980), which requires complex analytic normalization. As a consequence it holds that $N(X)$ is a $C$-semianalytic set of dimension $\leq\dim(X)-2$., Comment: 32 pages, 3 figures

We prove a Nullstellensatz result for global ideals of Denjoy-Carleman functions in both finitely generated and infinitely generated cases., Comment: 5 pages

It is shown that Denjoy-Carleman quasi-analytic rings of germs of functions in two or more variables fail to satisfy the Weierstrass Preparation Theorem. The result is proven via a non-extension theorem., Comment: 17 pages; accepted at the Advances in Mathematics

For an ideal of smooth functions that is either {\L}ojasiewicz or weakly {\L}ojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety in terms of the global {\L}ojasiewicz radical and Whitney closure. We also prove that the {\L}ojasiewicz radical of such an ideal is analytic-like in the sense that its saturation equals its Whitney closure. This allows us to recover in a different way Nullstellensatz results due to Bochnak and Adkins-Leahy and answer positively a modification of the Nullstellensatz conjecture due to Bochnak., Comment: 9 pages

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 \L ojasiewicz'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 $\widetilde{\sqrt[\text{\L}]{{\mathfrak a}}}$ of \L ojasiewicz's radical $\sqrt[\text{\L}]{{\mathfrak a}}$. 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 generalisation 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 neighbourhood 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 ${\mathcal Z}({\mathfrak p})$.