Write $\mathcal L(R)$ for the <polynomial length>. The <height bound for a polynomial evaluation> is
$$
H\!\left(\frac{P(\alpha_1,\ldots,\alpha_k)}
{Q(\alpha_1,\ldots,\alpha_k)}\right)
\leq
\max\{\mathcal L(P),\mathcal L(Q)\}
\prod_{j=1}^kH(\alpha_j)^{n_j},
$$
provided the denominator is nonzero. At non-Archimedean places the integral coefficients and <ultrametric inequality> give the local estimate without an extra constant; at Archimedean places the <triangle inequality> gives the polynomial length. Multiplication over every <place of a number field> and the <product formula> produce the displayed bound.

Taking $P(X,Y)=X+Y$ and $Q=1$ gives
$$
H(\alpha+\beta)\leq2H(\alpha)H(\beta).
$$
Taking $P(X)=X+1$ gives
$$
H(\alpha+1)\leq2H(\alpha).
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 166 / 2 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-166/2/e/solution