This question has likely an answer in some standard algebra textbook. If this is the case, I would like a reference.
Fix an integer $n$ and let $I \subseteq \mathbb{C}[x_0 , ... , x_n]$ be a homogeneous ideal in the polynomial ring. Suppose that $I$ is generated in degree at most $d$, namely $I = (f_1 , ... , f_k)$ with $f_j$'s homogeneous and $\deg(f_j) \leq d$.
Let $\mathfrak{m} = (x_0 , ... , x_n)$ be the maximal ideal generated by the variables and let $I_{sat} = I : \mathfrak{m}^\infty = \bigcup_{e} (I : \mathfrak{m}^e)$ be the saturation of $I$.
It is clear that there is a finite $E$ such that $I_{sat} = I : \mathfrak{m}^E$. This follows from the fact that $I_{sat}$ is finitely generated (by Noetherianity) and $\{ I : \mathfrak{m}^e : e \in \mathbb{N}\}$ form an ascending chain, which will stabilize when $e$ is large enough so that $I : \mathfrak{m}^e$ contains all the generators of $I_{sat}$.
My question is the following:
Is there an a priori upper bound on $E$ depending only on $n$ and $d$? What can one say about the worst possible $E$?
I can get silly upper bounds if I throw in also the number of generators (and in particular bounds only depending on $n$ and $d$ using that the generators are fewer than all polynomials of degree at most $d$). I am looking for something better than that.
Edit: I edited the title because I found out this number is called saturation number (see e.g. this unanswered question).
