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There is a notion of a Taylor Series for regular functions on an affine variety X, given a set of generators $\{u_1,\cdots,u_n\} $of the maximal ideal $\mathfrak{m}$ at a point $P$ that form a basis for the cotangent space $\mathfrak{m}/\mathfrak{m}^2$ we define the Taylor series of a function f as a formal series $F\in k[[T_1,\cdots,T_n]]$ such that the difference between the function and the truncated series lies in powers of the maximal ideal.

Shafarevich in Basic Algebraic Geometry I shows that any function in the local ring has a Taylor series, and if the point is non-singular then this Taylor series is unique.

He then asks as a exercise in the same chapter to show that, if the point is singular, then the Taylor series is never unique, i.e, there is an infinite number of Taylor series for any given function $f$.

I've solved the problem in the case that X is a hiper-surface simply by considering the generating polynomial as a power series, then tried to prove the result by induction on the codimension with no success.

How would one begin to show this result for arbitrary affine varieties?

David Melo
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    There is a natural map $\phi$ from $k[[T_1,\dots,T_n]]$ to the completion $\hat A$ of the local ring $A$ at $\frak m$, st $\phi(T_i)=u_i$. This map is surjective by Th. 8.4 in Matsumura, Commutative Algebra. Now $\hat A$ is regular iff $A$ is regular by https://stacks.math.columbia.edu/tag/07NY. Hence $\phi$ is not injective if $A$ is singular. – Damian Rössler Dec 01 '23 at 11:42
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    ... now if $F$ is a function in a neighbourhood of the point corresponding to $\frak m$, a Taylor series for $F$ is an element of $\phi^{-1}(F')$, where $F'$ is the image of $F$ in $A\subseteq \hat A$. So by the above, if $A$ is singular, there many choices of Taylor series. – Damian Rössler Dec 01 '23 at 11:43

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