= Solution
Choose a <number field> $L$ containing $\alpha_1,\ldots,\alpha_k$. At every place $v$ of $L$, put
$$
A_v=\prod_{j=1}^k\max(1,|\alpha_j|_v)^{n_j}.
$$
The <triangle inequality> gives
$$
\max\{|P(\boldsymbol\alpha)|_v,|Q(\boldsymbol\alpha)|_v\}
\leq c_vA_v,
$$
where $c_v=1$ at every non-Archimedean place because the coefficients are <integer>[integers], while at an Archimedean place one may take
$$
c_v=\max\{\mathcal L(P),\mathcal L(Q)\}.
$$
Raise these inequalities to the local weights and multiply over all places. The definition of the <Absolute multiplicative Weil height> and the <product formula> then give
$$
H\left(\frac{P(\alpha_1,\ldots,\alpha_k)}
{Q(\alpha_1,\ldots,\alpha_k)}\right)
\leq
\max\{\mathcal L(P),\mathcal L(Q)\}
\prod_{j=1}^kH(\alpha_j)^{n_j}.
$$
This is the <height bound for a polynomial evaluation>.
Solved by gpt-5.6-sol high.
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