About the problem
Update (2026-03-12): We have determined that any solution to this problem would not meet our bar of being a publishable result in its own right. As a result, we have removed the problem from the benchmark.
The problem consists of finding an explicit deformation from a specific curvilinear algebra to a specific monomial algebra. It is an instance of a more general problem, namely finding a general strategy for constructing such explicit deformations. It is this more general problem that meets the bar of being a publishable result in its own right. However, there is no reason to expect that solving this one instance would point the way to finding a general strategy.
Some problems in the benchmark have this character: solving a specific, verifiable instance very likely entails having solved a more general problem of broader interest. But, upon review, this problem does not have this character. We should have caught this before we included the problem in the benchmark, and we have amended our problem sourcing process to catch cases like this in the future.
In fact, this came to our attention after the problem was solved. The solution was generated by GPT-5.2 Pro running in a harness developed by David Turturean. The result still amounts to a highly non-trivial worked family of examples. Congratulations to David! Stay tuned for a preprint from David and the problem author, Gergely Berczi, describing this work.
Original Problem Summary: The study of deformations of Artinian algebras is central to the geometry of the Hilbert scheme of points, \(\mathrm{Hilb}^k(\mathbb{C}^n)\). The curvilinear component \(\mathrm{CHilb}_0^k(\mathbb{C}^n)\) plays a distinguished role: it parametrizes subschemes that are locally isomorphic to \(\mathrm{Spec}(k[t]/(t^k))\) inside \(\mathbb{C}^n\).
It is known that every such ideal is the deformation of the curvilinear ideal of the same dimension, but the proof is essentially nonconstructive. Such constructions would be useful for various applications. This problem is to provide an explicit deformation for a specific monomial algebra. The author expects that a solution to this problem would point toward a general strategy for constructing such deformations, which is the true target of this problem.
Warm-up: we ask for an explicit deformation to a simpler algebra, where the solution is known.
Attempts by AI
We have evaluated the following models on this problem. “Warm-up” refers to an easier variant of the problem with a known solution.
AI prompts
Warm-up
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}
Write a Python file using SymPy that defines the polynomial ring variables and the ideal generators.
It should define a function \texttt{solution} that defines SymPy symbols named
\texttt{x1},\dots,\texttt{xs} and \texttt{eps}, where $s\ge 3$. It should return 3 values:
\begin{itemize}
\item A list of generators using the SymPy symbols.
\item A list of all $s$ x-symbols in the same order used in the generators.
\item The symbol \texttt{eps} used in the generators.
\end{itemize}
Other contraints on the solution code:
\begin{itemize}
\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 code at the file level. You may include a "main" block for testing, but it will not be executed by the verifier.
\end{itemize}
An example is given below.
```
import sympy
def solution():
# Example: s = 4. Define x1, x2, x3, x4, and eps (epsilon)
x1, x2, x3, x4, eps = sympy.symbols("x1 x2 x3 x4 eps")
# List of x-symbols to return
x_symbols = [x1, x2, x3, x4]
# List of generators
i_generators = [
x1 * x2,
x1 * x3 - eps * x1**8,
x2 * x3 + eps * x2**8,
x4,
x1**8,
x2**8,
x3**8,
x4**8,
]
return i_generators, x_symbols, eps
``` Full problem
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}
Write a Python file using SymPy that defines the polynomial ring variables and the ideal generators.
It should define a function \texttt{solution} that defines SymPy symbols named
\texttt{x1},\dots,\texttt{xs} and \texttt{eps}, where $s\ge 3$. It should return 3 values:
\begin{itemize}
\item A list of generators using the SymPy symbols.
\item A list of all $s$ x-symbols in the same order used in the generators.
\item The symbol \texttt{eps} used in the generators.
\end{itemize}
Other contraints on the solution code:
\begin{itemize}
\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 code at the file level. You may include a "main" block for testing, but it will not be executed by the verifier.
\end{itemize}
An example is given below.
```
import sympy
def solution():
# Example: s = 4. Define x1, x2, x3, x4, and eps (epsilon)
x1, x2, x3, x4, eps = sympy.symbols("x1 x2 x3 x4 eps")
# List of x-symbols to return
x_symbols = [x1, x2, x3, x4]
# List of generators
i_generators = [
x1 * x2,
x1 * x3 - eps * x1**8,
x2 * x3 + eps * x2**8,
x4,
x1**8,
x2**8,
x3**8,
x4**8,
]
return i_generators, x_symbols, eps
```