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Let $R$ be a Noetherian ring and $P$ a prime ideal. Then the $n$-th symbolic power of $P$ is $$P^{(n)} = P^n R_P \cap R = \{ f \mid sf \in P^n \text{ for some } s \in R - P\}$$ (cf. wiki). We have $P^{(n)}$ is just the $P$-primary component of $P^n$.

Now, we consider $R = \mathbb{C}[X_1, \ldots, X_d]$ and $V = V(P)$ the variety defined by $P$. For each $a \in V$ we set $\mathfrak{m}_a$ the maximal ideal corresponding with $a$. We have $P = \cap_{a \in V} \mathfrak{m}_a$, since $R$ is a Jacobson ring. $P^{(n)}$ consists of functions with zeros of order $n$ along $V$.

Question 1. With above setting. Are the following equivalent:

  1. $f \in P^{(n)}$;
  2. $f \in \mathfrak{m}_a^n$ for all $a \in V$;
  3. $f, \frac{\partial^{\alpha} f}{\partial X^{\alpha}}$ vanish on $V$ for all $| \alpha | \le n-1$.

It is clear that $(2) \Leftrightarrow (3)$ and we can prove that $(1) \Rightarrow (3)$.

Question 2. Suppose Question has a positive answer. Can we have $(1) \Leftrightarrow (2)$ in more general rings (as Jacobson rings)?

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Question 1 is answered affirmatively in Eisenbud's "Commutative Algebra with a View Toward Algebraic Geometry" (p. 106, attributed to Zariski, Nagata). The text refers to "A Nullstellensatz with Nilpotents" [1979] by Eisenbud and Hochster for generalizations.

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