Explicit Deformations of Algebras

Work over a field $k$ of characteristic $0$. Construct an explicit flat one-parameter deformation from the algebra $k[t]/(t^3)$ to the algebra \[ A \;=\; k[x,y]/(x,y)^2. \] Equivalently, you must give an ideal $I \subset k[x_1,\dots,x_s,\epsilon]$ (for some $s\ge 2$) such that the family \[ \mathcal{A} \;=\; k[x_1,\dots,x_s,\epsilon]/I \] over $k[\epsilon]$ satisfies: \begin{enumerate} \item \textbf{Special fiber:} $\mathcal{A}\big|_{\epsilon=0} \cong A$. In particular, the specialization $I|_{\epsilon=0}$ must cut out an algebra isomorphic to $k[x,y]/(x,y)^2$ (it is allowed to use more generators $x_1,\dots,x_s$ as long as the fiber is isomorphic to $A$). \item \textbf{Generic fiber:} for $a \neq 0$, $\mathcal{A}\big|_{\epsilon=a} \cong k[t]/(t^3)$. \end{enumerate} \subsection*{Mandatory output format} Submit a Python/SymPy snippet that defines the polynomial ring variables and the ideal generators. \begin{itemize} \item You may choose any $s\ge 2$. Use variables named \texttt{x1},\dots,\texttt{xs} and \texttt{eps}. \item Define \texttt{x\_vars} as the tuple $(x_1,\dots,x_s)$ in the same order used in \texttt{I\_gens}. \item Provide the ideal generators as a Python list named \texttt{I\_gens} of SymPy expressions in the symbols \texttt{x1},\dots,\texttt{xs},\texttt{eps}. \item Coefficients must be integers only (no fractions, no floats, no extra parameters besides \texttt{eps}). \item Use only polynomial operations \texttt{+}, \texttt{-}, \texttt{*}, \texttt{**}; do not use division. \item Do not include any explanatory text outside the code block. \end{itemize} \subsection*{Exact skeleton you must output (fill in $s$ and \texttt{I\_gens}).} \begin{verbatim} import sympy as sp # choose s >= 2 x1, x2, eps = sp.symbols("x1 x2 eps") # extend as needed: x3, x4, ... x_vars = (x1, x2) # extend consistently I_gens = [ # polynomial generators in x1,x2,(x3,x4,...),eps with integer coefficients ] \end{verbatim}

Work over a field $k$ of characteristic $0$. Construct an explicit flat one-parameter deformation from the curvilinear algebra $k[t]/(t^{22})$ to the monomial "spider" algebra \[ A \;=\; k[x,y,z]/(x^8,\;y^8,\;z^8,\;xy,\;xz,\;yz). \] Equivalently, you must give an ideal $I \subset k[x_1,\dots,x_s,\epsilon]$ (for some $s\ge 3$) such that the family \[ \mathcal{A} \;=\; k[x_1,\dots,x_s,\epsilon]/I \] over $k[\epsilon]$ satisfies: \begin{enumerate} \item \textbf{Special fiber:} $\mathcal{A}\big|_{\epsilon=0} \cong A$. In particular, the specialization $I|_{\epsilon=0}$ must cut out an algebra isomorphic to $k[x,y,z]/(x^8,y^8,z^8,xy,xz,yz)$ (it is allowed to use more generators $x_1,\dots,x_s$ as long as the fiber is isomorphic to $A$). \item \textbf{Generic fiber:} for $a \neq 0$, $\mathcal{A}\big|_{\epsilon=a} \cong k[t]/(t^{22})$. \end{enumerate} \subsection*{Mandatory output format} Submit a Python/SymPy snippet that defines the polynomial ring variables and the ideal generators. \begin{itemize} \item You may choose any $s\ge 3$. Use variables named \texttt{x1},\dots,\texttt{xs} and \texttt{eps}. \item Define \texttt{x\_vars} as the tuple $(x_1,\dots,x_s)$ in the same order used in \texttt{I\_gens}. \item Provide the ideal generators as a Python list named \texttt{I\_gens} of SymPy expressions in the symbols \texttt{x1},\dots,\texttt{xs},\texttt{eps}. \item Coefficients must be integers only (no fractions, no floats, no extra parameters besides \texttt{eps}). \item Use only polynomial operations \texttt{+}, \texttt{-}, \texttt{*}, \texttt{**}; do not use division. \item Do not include any explanatory text outside the code block. \end{itemize} \subsection*{Exact skeleton you must output (fill in $s$ and \texttt{I\_gens}).} \begin{verbatim} import sympy as sp # choose s >= 3 x1, x2, x3, eps = sp.symbols("x1 x2 x3 eps") # extend as needed: x4, x5, ... x_vars = (x1, x2, x3) # extend consistently I_gens = [ # polynomial generators in x1,x2,x3,(x4,...),eps with integer coefficients ] \end{verbatim}