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There is a paper by Hans-Jörg Reiffen called "Fastholomorphe Algebren" (1970) where he apparently proves something like this. Here is a link to the springer publication. I have not read the paper in detail so I can not really comment on the methods used there. Moreover, the paper is written entirely in german. The last highlighted statement in the appendix reads (translated to english by me):

"Let $G$ be a domain in $\mathbb{C}^N$ and $X\subseteq G$ a real analytic subset, which in the points of an open dense subset of $X$ is complex analytic, then $X$ is complex analytic."

EDIT: Below I have translated the statement 1' and 2' from the appendix of the mentioned paper. Apparently the above quoted statement is a corollary from these two facts, at least that is what the author of the paper states. In the paper the notation $R_{\omega}$ denotes the ring of complex-valued real analytic functions.

Satz 1'. Let $X_0$ be a germ of an irreducible real analytic set in $\mathbb{R}^N$. Then there exists a real analytic subgerm $S_0$ of $X_0$ with the following properties:

(a) $\mathrm{dim}X_0> \mathrm{dim}S_0$

(b) There exist representatives $X$ of $X_0$ and $S$ of $S_0$ in an open neighbourhood $U$ of $0$ such that for all $x\in X\setminus S$ the germ $X_x$ is non-singular and $\mathrm{dim}X_x= \mathrm{dim}S_0$.

(c) If $f_1,...,f_n$ is a generating system of $J_{\omega}(X_0):=\left\{f\in R_{\omega} : f\mid_{X_0}=0\right\}$, the $f_1,....,f_n$ is a coherence basis on $X_0\setminus S_0$ (i.e. there exists an open neighbourhood $U$ of $0$ with representatives $X$ of $X_0$ and $S$ of $S_0$, such that the $f_1,....,f_n$ are defined on $U$ and for all $x\in X\setminus S$ the germs $(f_1)_x,...,(f_n)_x$ generate the ideal $J_{\omega}(X_x)$.)

Satz 2'. Let $X_0$ be a germ of an irreducible real analytic set in the origin of $\mathbb{C}^N$. Further assume that there exists a set $Y$ in a representative $X$ of $X_0$ with the following properties:

(a) $0$ is an accumulation point of $Y$.

(b) For $y\in Y$ one has $\mathrm{dim}X_y=\mathrm{dim}X_0$ and $X_y$ is complex analytic.

Then $X_0$ is complex analytic.

There is a paper by Hans-Jörg Reiffen called "Fastholomorphe Algebren" (1970) where he apparently proves something like this. Here is a link to the springer publication. I have not read the paper in detail so I can not really comment on the methods used there. Moreover, the paper is written entirely in german. The last highlighted statement in the appendix reads (translated to english by me):

"Let $G$ be a domain in $\mathbb{C}^N$ and $X\subseteq G$ a real analytic subset, which in the points of an open dense subset of $X$ is complex analytic, then $X$ is complex analytic."

EDIT: Below I have translated the statement 1' and 2' from the appendix of the mentioned paper. Apparently the above quoted statement is a corollary from these two facts, at least that is what the author of the paper states. In the paper the notation $R_{\omega}$ denotes the ring of complex-valued real analytic functions.

Satz 1'. Let $X_0$ be a germ of an irreducible real analytic set in $\mathbb{R}^N$. Then there exists a real analytic subgerm $S_0$ of $X_0$ with the following properties:

(a) $\mathrm{dim}X_0> \mathrm{dim}S_0$

(b) There exist representatives $X$ of $X_0$ and $S$ of $S_0$ in an open neighbourhood $U$ of $0$ such that for all $x\in X\setminus S$ the germ $X_x$ is non-singular and $\mathrm{dim}X_x= \mathrm{dim}S_0$.

(c) If $f_1,...,f_n$ is a generating system of $J_{\omega}(X_0):=\left\{f\in R_{\omega} : f\mid_{X_0}=0\right\}$, then the $f_1,....,f_n$ is a coherence basis on $X_0\setminus S_0$ (i.e. there exists an open neighbourhood $U$ of $0$ with representatives $X$ of $X_0$ and $S$ of $S_0$, such that the $f_1,....,f_n$ are defined on $U$ and for all $x\in X\setminus S$ the germs $(f_1)_x,...,(f_n)_x$ generate the ideal $J_{\omega}(X_x)$.)

Satz 2'. Let $X_0$ be a germ of an irreducible real analytic set in the origin of $\mathbb{C}^N$. Further assume that there exists a set $Y$ in a representative $X$ of $X_0$ with the following properties:

(a) $0$ is an accumulation point of $Y$.

(b) For $y\in Y$ one has $\mathrm{dim}X_y=\mathrm{dim}X_0$ and $X_y$ is complex analytic.

Then $X_0$ is complex analytic.

There is a paper by Hans-Jörg Reiffen called "Fastholomorphe Algebren" (1970) where he apparently proves something like this. Here is a link to the springer publication. I have not read the paper in detail so I can not really comment on the methods used there. Moreover, the paper is written entirely in german. The last highlighted statement in the appendix reads (translated to english by me):

"Let $G$ be a domain in $\mathbb{C}^N$ and $X\subseteq G$ a real analytic subset, which in the points of an open dense subset of $X$ is complex analytic, then $X$ is complex analytic."

EDIT: Below I have translated the statement 1' and 2' from the appendix of the mentioned paper. Apparently the above quoted statement is a corollary from these two facts, at least that is what the author of the paper states. In the paper the notation $R_{\omega}$ denotes the ring of complex-valued real analytic functions.

Satz 1'. Let $X_0$ be a germ of an irreducible real analytic set in $\mathbb{R}^N$. Then there exists a real analytic subgerm $S_0$ of $X_0$ with the following properties:

(a) $\mathrm{dim}X_0> \mathrm{dim}S_0$

(b) There exist representatives $X$ of $X_0$ and $S$ of $S_0$ in an open neighbourhood $U$ of $0$ such that for all $x\in X\setminus S$ the germ $X_x$ is non-singular and $\mathrm{dim}X_x= \mathrm{dim}S_0$.

(c) If $f_1,...,f_n$ is a generating system of $J_{\omega}(X_0):=\left\{f\in R_{\omega} : f\mid_{X_0}\right\}$, the $f_1,....,f_n$ is a coherence basis on $X_0\setminus S_0$ (i.e. there exists an open neighbourhood $U$ of $0$ with representatives $X$ of $X_0$ and $S$ of $S_0$, such that the $f_1,....,f_n$ are defined on $U$ and for all $x\in X\setminus S$ the germs $(f_1)_x,...,(f_n)_x$ generate the ideal $J_{\omega}(X_x)$.)

Satz 2'. Let $X_0$ be a germ of an irreducible real analytic set in the origin of $\mathbb{C}^N$. Further assume that there exists a set $Y$ in a representative $X$ of $X_0$ with the following properties:

(a) $0$ is an accumulation point of $Y$.

(b) For $y\in Y$ one has $\mathrm{dim}X_y=\mathrm{dim}X_0$ and $X_y$ is complex analytic.

Then $X_0$ is complex analytic.

There is a paper by Hans-Jörg Reiffen called "Fastholomorphe Algebren" (1970) where he apparently proves something like this. Here is a link to the springer publication. I have not read the paper in detail so I can not really comment on the methods used there. Moreover, the paper is written entirely in german. The last highlighted statement in the appendix reads (translated to english by me):

"Let $G$ be a domain in $\mathbb{C}^N$ and $X\subseteq G$ a real analytic subset, which in the points of an open dense subset of $X$ is complex analytic, then $X$ is complex analytic."

EDIT: Below I have translated the statement 1' and 2' from the appendix of the mentioned paper. Apparently the above quoted statement is a corollary from these two facts, at least that is what the author of the paper states. In the paper the notation $R_{\omega}$ denotes the ring of complex-valued real analytic functions.

Satz 1'. Let $X_0$ be a germ of an irreducible real analytic set in $\mathbb{R}^N$. Then there exists a real analytic subgerm $S_0$ of $X_0$ with the following properties:

(a) $\mathrm{dim}X_0> \mathrm{dim}S_0$

(b) There exist representatives $X$ of $X_0$ and $S$ of $S_0$ in an open neighbourhood $U$ of $0$ such that for all $x\in X\setminus S$ the germ $X_x$ is non-singular and $\mathrm{dim}X_x= \mathrm{dim}S_0$.

(c) If $f_1,...,f_n$ is a generating system of $J_{\omega}(X_0):=\left\{f\in R_{\omega} : f\mid_{X_0}=0\right\}$, the $f_1,....,f_n$ is a coherence basis on $X_0\setminus S_0$ (i.e. there exists an open neighbourhood $U$ of $0$ with representatives $X$ of $X_0$ and $S$ of $S_0$, such that the $f_1,....,f_n$ are defined on $U$ and for all $x\in X\setminus S$ the germs $(f_1)_x,...,(f_n)_x$ generate the ideal $J_{\omega}(X_x)$.)

Satz 2'. Let $X_0$ be a germ of an irreducible real analytic set in the origin of $\mathbb{C}^N$. Further assume that there exists a set $Y$ in a representative $X$ of $X_0$ with the following properties:

(a) $0$ is an accumulation point of $Y$.

(b) For $y\in Y$ one has $\mathrm{dim}X_y=\mathrm{dim}X_0$ and $X_y$ is complex analytic.

Then $X_0$ is complex analytic.

There is a paper by Hans-Jörg Reiffen called "Fastholomorphe Algebren" (1970) where he apparently proves something like this. Here is a link to the springer publication. I have not read the paper in detail so I can not really comment on the methods used there. Moreover, the paper is written entirely in german. The last highlighted statement in the appendix reads (translated to english by me):

"Let $G$ be a domain in $\mathbb{C}^N$ and $X\subseteq G$ a real analytic subset, which in the points of an open dense subset of $X$ is complex analytic, then $X$ is complex analytic."

EDIT: Below I have translated the statement 1' and 2' from the appendix of the mentioned paper. Apparently the above quoted statement is a corollary from these two facts, at least that is what the author of the paper states. In the paper the notation $R_{\omega}$ denotes the ring of complex-valued real analytic functions.

Satz 1'. Let $X_0$ be a germ of an irreducible real analytic set in $\mathbb{R}^N$. Then there exists a real analytic subgerm $S_0$ of $X_0$ with the following properties:

(a) $\mathrm{dim}X_0> \mathrm{dim}S_0$

(b) There exist representatives $X$ of $X_0$ and $S$ of $S_0$ in an open neighbourhood $U$ of $0$ such that for all $x\in X\setminus S$ the germ $X_x$ is non-singular and $\mathrm{dim}X_x= \mathrm{dim}S_0$.

(c) If $f_1,...,f_n$ is a generating system of $J_{\omega}(X_0):=\left\{f\in R_{\omega} : f\mid_{X_0}\right\}$, the $f_1,....,f_n$ is a coherence basis on $X_0\setminus S_0$ (i.e. there exists an open neighbourhood $U$ of $0$ with representatives $X$ of $X_0$ and $S$ of $S_0$, such that the $f_1,....,f_n$ are defined on $U$ and for all $x\in X\setminus S$ the germs $(f_1)_x,...,(f_n)_x$ generate the ideal $J_{\omega}(X_x)$.)

Satz 2'. Let $X_0$ be a germ of an irreducible real analytic set in the origin of $\mathbb{C}^N$. Further, there exists a set $Y$ in a representative $X$ of $X_0$ with the following properties:

(a) $0$ is an accumulation point of $Y$.

(b) For $y\in Y$ one has $\mathrm{dim}X_y=\mathrm{dim}X_0$ and $X_y$ is complex analytic.

Then $X_0$ is complex analytic.

There is a paper by Hans-Jörg Reiffen called "Fastholomorphe Algebren" (1970) where he apparently proves something like this. Here is a link to the springer publication. I have not read the paper in detail so I can not really comment on the methods used there. Moreover, the paper is written entirely in german. The last highlighted statement in the appendix reads (translated to english by me):

"Let $G$ be a domain in $\mathbb{C}^N$ and $X\subseteq G$ a real analytic subset, which in the points of an open dense subset of $X$ is complex analytic, then $X$ is complex analytic."

EDIT: Below I have translated the statement 1' and 2' from the appendix of the mentioned paper. Apparently the above quoted statement is a corollary from these two facts, at least that is what the author of the paper states. In the paper the notation $R_{\omega}$ denotes the ring of complex-valued real analytic functions.

Satz 1'. Let $X_0$ be a germ of an irreducible real analytic set in $\mathbb{R}^N$. Then there exists a real analytic subgerm $S_0$ of $X_0$ with the following properties:

(a) $\mathrm{dim}X_0> \mathrm{dim}S_0$

(b) There exist representatives $X$ of $X_0$ and $S$ of $S_0$ in an open neighbourhood $U$ of $0$ such that for all $x\in X\setminus S$ the germ $X_x$ is non-singular and $\mathrm{dim}X_x= \mathrm{dim}S_0$.

(c) If $f_1,...,f_n$ is a generating system of $J_{\omega}(X_0):=\left\{f\in R_{\omega} : f\mid_{X_0}\right\}$, the $f_1,....,f_n$ is a coherence basis on $X_0\setminus S_0$ (i.e. there exists an open neighbourhood $U$ of $0$ with representatives $X$ of $X_0$ and $S$ of $S_0$, such that the $f_1,....,f_n$ are defined on $U$ and for all $x\in X\setminus S$ the germs $(f_1)_x,...,(f_n)_x$ generate the ideal $J_{\omega}(X_x)$.)

Satz 2'. Let $X_0$ be a germ of an irreducible real analytic set in the origin of $\mathbb{C}^N$. Further assume that there exists a set $Y$ in a representative $X$ of $X_0$ with the following properties:

(a) $0$ is an accumulation point of $Y$.

(b) For $y\in Y$ one has $\mathrm{dim}X_y=\mathrm{dim}X_0$ and $X_y$ is complex analytic.

Then $X_0$ is complex analytic.