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