If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? such that
$S^m \in (X^a - S^b, X^d +X^{d-1}S_1 + X^{d-2}S^2 + \ldots + S^d)$.
Given two polynomial $f(X,S), g(X,S)$, is there a general formula in terms of some data of $f(X,S), g(X,S)$ to compute the minimal number $m$ such that $S^m \in {\frak a}$?
Pierre
There is certainly some structure in your example, so maybe also to other ideals that you have in mind?
The first thing I would do is to make experiments and try to guess a formula. Here is Macaulay2 code for this:
R = QQ[X,S]
f = (a,b,d) -> ideal (X^a - S^b, sum flatten entries basis (d,R))
The first line creates a ring, and the second line defines a function that, given a,b,d, returns the ideal in your example. Now we can fix a=b=2, say and look at this ideal as a function of d. I would suggest to compute its primary decomposition directly. This gives you a lot of information. If you do that, you'll observe that the ideal contains a power of S if and only if d is even and that the power seems to be d+1. Now you can try to prove this from the generators... If a=b=3, then it seems to contain a power of S if and only if d leaves remainder 0 or 1 mod 3. The power seems to be d+2 in this case.
I hope you can see how useful experimentation is for such questions. Maybe after doing this kind of practice for a couple of ideals you can come up with an intuition what properties of $f$ and $g$ you want to look at.
Do you assume the ideal is homogeneous? No power of $S$ belong to $(X,S-1)$.
If you assume that the ideal is homogeneous, then $S^m$ is in the ideal $(f,g)$ for $m=\deg f+\deg g-1$. Indeed all homogeneous polynomials of that degree are in the ideal $(f,g)$. This is a simple Hilbert series computation, just use the fact that the Koszul complex resolves $K[X,S]/(f,g)$ and use it to compute the Hilbert series of $K[X,S]/(f,g)$.
