 Solution

For $P=[x_0:\cdots :x_N]\in {\mathbb{P}}^N(\mathbb{Q})$, choose coprime integer coordinates and define the projective heightWrite the morphism $F:{\mathbb{P}}^1\to {\mathbb{P}}^1$ as $F=[F_0:F_1]$, where $F_0,F_1\in \mathbb{Z}[X,Y]$ are homogeneous of degree $d$ with no common zero. Bounding each polynomial by the sum of the absolute values of its coefficients gives

Because $F_0$ and $F_1$ have no common projective zero, the Projective Nullstellensatz gives an integer $m\ge d$ and homogeneous polynomials $A_{ij}$ such that suitable nonzero integer multiples of $X^m$ and $Y^m$ lie in the ideal $(F_0,F_1)$. Evaluating at primitive coordinates and using the same coefficient bound givesafter absorbing the bounded common divisor of $F_0(x,y)$ and $F_1(x,y)$ into $C$. ThereforeThis is height growth under a morphism of the projective line.