Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (<https://stacks.math.columbia.edu/tag/07DR>). Then, $R$ is the homomorphic image of a localization of some polynomial ring over $k$.


My question is: Does there exist a surjective $k$-algebra homomorphism $k[X\_1,\ldots, X\_n]\_P \to R$, for some integer $n\geq 0$ and prime ideal $P $ of $k[X\_1,\ldots, X\_n]$, such that $\mu(\mathfrak m)=\dim k[X\_1,\ldots, X\_n]\_P $?


(Here, $\mu(\mathfrak m)$ is the minimal number of generators of $\mathfrak m$).


Some obvious thoughts: I think it is enough to show that there exists integer $n\geq 0$ and prime ideal $P $ of $k[X\_1,\ldots, X\_n]$ such that $R \cong \dfrac{ k[X\_1,\ldots, X\_n]\_P}{I}$ for some ideal $I$ contained in the square of the maximal ideal of $k[X\_1,\ldots, X\_n]\_P$.



On "minimal presentation" of local rings essentially of finite type over a field
If $R$ is the local ring of the point $P$ on a smooth $n$-dimensional $k$-variety $V$ then any such homomorphism would be an isomorphism. So it cannot exist if $V$ is irrational. For example, take $V$ to be an elliptic curve.


