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Let $I\subseteq{\mathbb C}[X_1,\dotsc,X_n]$ be an ideal, and

let $V\subseteq{\mathbb C}^n$ be the corresponding algebraic set

($V$ consists of those $x$ at which all $f\in I$ vanish).

Is it true that then there exists an integer $N$ such that:

if a function $f\in{\mathbb C}[X_1,\dotsc,X_n]$ and all its partial derivatives up to order $N$ vanish on $V$,

then $f\in I$?

MOTIVATION. I believe this would imply an affirmative answer to my question about commutative algebras and representations of the category of finite sets:

Commutative algebras and Gamma-modules

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3 Answers 3

up vote 9 down vote accepted

The answer is yes. Let $I = q_1 \cap q_2 \cap \cdots \cap q_k$ be the primary decomposition of $I$. Let $p_i$ be the radical $\sqrt{q_i}$ and let $N_i$ be large enough that $q_i \supseteq p_i^{N_i}$. I claim that we can take $N= \max(N_i)$. We'll abbreviate $\mathbb{C}[x_1, \ldots, x_n]$ to $A$.

Let $f$ be a polynomial as in the statement of the question. In order to show that $f \in I$, it is enough to show that $f \in q_i$ for every $i$. We focus on one $q$ to pay attention to, so we can drop the subscript $i$ and just talk about $p$, $q$ and $N$. Let $W$ be the variety of $p$.

Now, $A$ is regular, so $A_p$ is regular. In other words, if $p$ has codimension $d$, there are transverse smooth hypersurfaces $y_1$, $y_2$, ..., $y_d$ which generate the maximal ideal of $A_p$. So there is some $u \not \in p$ such that $u^{-1} p = \langle y_1, \ldots, y_d \rangle$ in the localization $A[u^{-1}]$.

We claim that $f$, as an element of $A[u^{-1}]$ is in $u^{-1} p^N$. Proof: Write $$f=\sum_{i_1+\cdots+i_d < N} b_{i_1 \ldots i_d} y_1^{i_1} \cdots y_d^{i_d} + r$$ with $r \in u^{-1} p^N$ and with all the $b$'s either $0$ or not in $p$. We want to show the $b$'s are zero. If not, let $(i_1, \ldots, i_d)$ be minimal with the corresponding $b$ nonzero. Since $b$ is not in $p$, it is not zero on $W$. Choose $w$ in $W$ where neither $b$ nor $u$ vanishes.

Since the $y_i$'s are transverse hypersurfaces, we can take $\sum i_j$ derivatives to obtain a polynomial with leading term $\mbox{nonzero stuff} \cdot \prod (i_j)! \cdot b$ at $w$. Since we are in characteristic zero, this doesn't vanish at $w$. (In finite characteristic, the factorials might be zero.) But $w \in W \subseteq V$, so this derivative is supposed to vanish at $w$. This contradiction shows that $f \in u^{-1} p^N$.

Since $q \supseteq p^N$, we have $f \in u^{-1} q$ so $u^k f \in q$ for some $k$. But $q$ is primary and $u \not \in p = \sqrt{q}$. So this shows $f \in q$, as desired.

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Thank you very much! –  Semen Podkorytov Nov 12 '10 at 10:22
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The answer "yes" follows immediately from Corollary 2 in "A Nullstellensatz with Nilpotents and Zariski’s Main Lemma on Holomorphic Functions" by D. Eisenbud and M. Hochster, J. Algebra 58 (1979), 157-161, http://deepblue.lib.umich.edu/bitstream/2027.42/23570/1/0000531.pdf

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I think that the answer is 'yes'. You should estimate how 'far' I is from its nilradical (for which one can take $N=0$). Since the ring of polynomials is Noetherian, you could obtain the nilradical of I by adding a finite number of roots of elements of I to it (i.e. you obtain some $I_1$, add some root of its element, get $I_2$, etc.). I strongly suspect that you can get some value of $N$ looking at this process.

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