Lemma 33.45.4. In the situation of Definition 33.45.3 the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ is an integer.

Proof. Any numerical polynomial of degree $e$ in $n_1, \ldots , n_ d$ can be written uniquely as a $\mathbf{Z}$-linear combination of the functions ${n_1 \choose k_1}{n_2 \choose k_2} \ldots {n_ d \choose k_ d}$ with $k_1 + \ldots + k_ d \leq e$. Apply this with $e = d$. Left as an exercise. $\square$

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