I've got a question about Fulton's Intersection Theory, Example 15.3.1.
Let $f: X \to Y$ be a closed imbedding of non-singular varieties with $\operatorname{codim}(X;Y) = d$. Then Riemann-Roch without Denominators is $$c(f_* E) = 1 + f_* P(N,E),$$ where $E$ is a vector bundle of rank $e$, $N$ is the normal bundle of $X$ in $Y$ and $P$ is the unique polynomial in the Chern classes of $N,E$ such that for all varieties $V$ and vector bundles $D,E$ on $V$ of rank $d,e$ one has $$c(\Lambda^* D^\vee \otimes E) - 1 = c_d(D) \cdot P(D,E).$$
In Example 15.3.1 Fulton claims $$P_0(N,E) = (-1)^{d-1} (d-1)! \,e,$$ where $P_0$ is the degree zero term of $P$. I don't see how to arrive at this equality.
Fulton also references Example 15.2.1 (c), which asserts that if $c_q(E) = 0$ for all $0 < q < d$, then $$x_1^d + \cdots + x_e^d =: p_d = (-1)^{d-1} d \cdot c_d(E),$$ where $x_1, \dotsc, x_e$ are the Chern roots of $E$. This assumption is certainly true for $f_*E$, because $c_q(f_* E) = f_* P_{q-d}(N,E)$ for $q > 0$. But I don't see any connection between $P_0$ and $p_d$, so I don't know how to apply this.
I also verified the equation in question for $d = 2, e = 1$.