Let $X$ be the hyper-surface defined by
$$f:=\sum\_{i=1}^k x\_i^n=0$$
in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal
$$I=(x\_1^{n-1},\dots , x\_k^{n-1}) $$ 
What is the multiplicity of $Y$ in $X$? Is it $n(n-1)^{k-1}$?


thanks



Samuel multiplicity
This follows from the discussion in Fulton's book on intersection theory, Example 4.3.5. However, your terminology isn't quite right. What you want is the *multiplicity of $Y$ along $X$ at $0$*, see Fulton, Example 4.3.4. Fulton quotes a result of Samuel that one can reduce this multiplicity to the multiplicity for an ideal $I' \subset I$ generated by a regular sequence $f\_1, \ldots, f\_{k - 1}$ where the $f\_i$ are linear combinations of the generators you gave for $I$ (this is actually a bit surprising). Then $I'$ defines a complete intersection supported at $0$ given by homogeneous polynomials of degrees $n, n - 1, \ldots, n - 1$. A standard argument shows that here the number is indeed as you predict.


