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Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let

\begin{equation*} T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\![X_1,\ldots, X_n ]\!] \mid a_{\nu_1, \ldots, \nu_n } \rightarrow 0 \text{ for } \sum \nu_i\rightarrow \infty \} \end{equation*}

be the Tate algebra. For $f=\sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in T_n$ the Gauss norm $|\, \,\, |$ ist defined by $|f|=\max_{(\nu_1, \ldots, \nu_n)} |a_{\nu_1, \ldots, \nu_n }|$. Then $T_n$ is complete with respect to $|\, \,\, |$ and $K[X_1,\ldots, X_n]$ is dense in it.

For any (closed) ideal $\mathfrak{a} \subset T_n $, we have for $\overline{f} \in T_n/\mathfrak{a}$ the residue norm defined by $|\overline{f}|_{res}=\inf \{|h| \mid h \in \overline{f}\}$. Then $T_n/\mathfrak{a}$ complete with respect to the residue norm.

Now let $I \subset K[X_1,\ldots, X_n]$ be closed with respect to the Gauß norm. Then the residue semi-norm on $K[X_1,\ldots, X_n]/I$ is actually a norm.

Then I have the following question.

1. Are there Ideals $I \subset K[X_1,\ldots, X_n]$ which are non-closed?
2. Is the completion of $I$ with respect to the Gauss norm $I\cdot T_n$?
3. Is the completion of $K[X_1,\ldots, X_n]/I$ with respect to the residue norm $T_n/(I\cdot T_n)$?

Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let

\begin{equation*} T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\![X_1,\ldots, X_n ]\!] \mid a_{\nu_1, \ldots, \nu_n } \rightarrow 0 \text{ for } \sum \nu_i\rightarrow \infty \} \end{equation*}

be the Tate algebra. For $f=\sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in T_n$ the Gauss norm $|\, \,\, |$ ist defined by $|f|=\max_{(\nu_1, \ldots, \nu_n)} |a_{\nu_1, \ldots, \nu_n }|$. Then $T_n$ is complete with respect to $|\, \,\, |$ and $K[X_1,\ldots, X_n]$ is dense in it.

For any (closed) ideal $\mathfrak{a} \subset T_n $, we have for $\overline{f} \in T_n/\mathfrak{a}$ the residue norm defined by $|\overline{f}|_{res}=\inf \{|h| \mid h \in \overline{f}\}$. Then $T_n/\mathfrak{a}$ complete with respect to the residue norm.

Now let $I \subset K[X_1,\ldots, X_n]$ be closed with respect to the Gauß norm. Then the residue semi-norm on $K[X_1,\ldots, X_n]/I$ is actually a norm.

Then I have the following question.

1. Are there ideals $I \subset K[X_1,\ldots, X_n]$ which are non-closed resp. how do closed ideals look like?
2. Is the completion of $I$ with respect to the Gauss norm $I\cdot T_n$?
3. Is the completion of $K[X_1,\ldots, X_n]/I$ with respect to the residue norm $T_n/(I\cdot T_n)$?

Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let

\begin{equation*} T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\![X_1,\ldots, X_n ]\!] \mid a_{\nu_1, \ldots, \nu_n } \rightarrow 0 \text{ for } \sum \nu_i\rightarrow \infty \} \end{equation*}

be the Tate algebra. For $f=\sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in T_n$ the Gauss norm $|\, \,\, |$ ist defined by $|f|=\max_{(\nu_1, \ldots, \nu_n)} |a_{\nu_1, \ldots, \nu_n }|$. Then $T_n$ is complete with respect to $|\, \,\, |$ and $K[X_1,\ldots, X_n]$ is dense in it.

For any (closed) ideal $\mathfrak{a} \subset T_n $, we have for $\overline{f} \in T_n/\mathfrak{a}$ the residue norm defined by $|\overline{f}|_{res}=\inf \{|h| \mid h \in \overline{f}\}$. Then $T_n/\mathfrak{a}$ complete with respect to the residue norm.

Now let $I \subset K[X_1,\ldots, X_n]$ be closed with respect to the Gauß norm. Then the residue semi-norm on $K[X_1,\ldots, X_n]/I$ is actually a norm.

Then I have the following question.

1. Are there Ideals which are non-closed?
2. Is the completion of $I$ with respect to the Gauss norm $I\cdot T_n$?
3. Is the completion of $K[X_1,\ldots, X_n]/I$ with respect to the residue norm $T_n/(I\cdot T_n)$?

Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let

\begin{equation*} T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\![X_1,\ldots, X_n ]\!] \mid a_{\nu_1, \ldots, \nu_n } \rightarrow 0 \text{ for } \sum \nu_i\rightarrow \infty \} \end{equation*}

be the Tate algebra. For $f=\sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in T_n$ the Gauss norm $|\, \,\, |$ ist defined by $|f|=\max_{(\nu_1, \ldots, \nu_n)} |a_{\nu_1, \ldots, \nu_n }|$. Then $T_n$ is complete with respect to $|\, \,\, |$ and $K[X_1,\ldots, X_n]$ is dense in it.

For any (closed) ideal $\mathfrak{a} \subset T_n $, we have for $\overline{f} \in T_n/\mathfrak{a}$ the residue norm defined by $|\overline{f}|_{res}=\inf \{|h| \mid h \in \overline{f}\}$. Then $T_n/\mathfrak{a}$ complete with respect to the residue norm.

Now let $I \subset K[X_1,\ldots, X_n]$ be closed with respect to the Gauß norm. Then the residue semi-norm on $K[X_1,\ldots, X_n]/I$ is actually a norm.

Then I have the following question.

1. Are there Ideals $I \subset K[X_1,\ldots, X_n]$ which are non-closed?
2. Is the completion of $I$ with respect to the Gauss norm $I\cdot T_n$?
3. Is the completion of $K[X_1,\ldots, X_n]/I$ with respect to the residue norm $T_n/(I\cdot T_n)$?