= Height bound for a polynomial evaluation
Let $P,Q\in\mathbb Z[X_1,\ldots,X_k]$ have degree at most $n_j$ in $X_j$. Then
$$
H\left(\frac{P(\alpha_1,\ldots,\alpha_k)}
{Q(\alpha_1,\ldots,\alpha_k)}\right)
\leq
\max\{\mathcal L(P),\mathcal L(Q)\}
\prod_{j=1}^kH(\alpha_j)^{n_j}
$$
whenever the quotient is defined. At each non-Archimedean place, the integral coefficients contribute at most one; at the Archimedean places, the <triangle inequality> contributes the <polynomial length>. Multiplying these local estimates and using the <product formula> gives the claim.
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