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](https://en.wikipedia.org/wiki/Symbolic_power_of_a_prime_ideal)). 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)?



a question on symbolic power
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.


