= Solution
For $P=[x:y]\in\mathbb P^1(\mathbb Q)$ represented by coprime integers, its <naive height on the projective line> is
$$
H(P)=\max\{|x|,|y|\}.
$$
Write the degree-$d$ morphism as $\Phi=[F:G]$, where $F,G\in\mathbb Z[X,Y]$ are homogeneous of degree $d$ with no common projective zero. Bounding their coefficients gives
$$
H(\Phi(P))\leq c_2H(P)^d.
$$
For the reverse inequality, the nonvanishing of the <resultant> of $F$ and $G$ gives homogeneous Bézout identities expressing fixed nonzero integer multiples of powers of $X$ and $Y$ as polynomial combinations of $F$ and $G$. Evaluating at $(x,y)$, removing the common divisor of $F(x,y)$ and $G(x,y)$, and taking the larger of $|x|,|y|$ gives
$$
H(P)^d\leq C H(\Phi(P)).
$$
Thus $c_1H(P)^d\leq H(\Phi(P))\leq c_2H(P)^d$ for constants depending only on $\Phi$.
Back to article page
