Let $\phi \in K(z)$ be a rational function of degree $d$ over a number field $K$ and write $\phi = f/g$ for coprime $f,g \in K[z]$. Then, there exist positive constants $c_1, c_2$ such that $c_1 H(\alpha)^d \leq H(\phi(\alpha)) \leq c_2 H(\alpha)^d$ for all $\alpha \in \overline{K}$.
Problem 3.8 in Silverman's text relates to finding explicit forms of $c_1, c_2$ that depend only upon the height of $\phi$ and the degree $d$. My question concerns the lower bound. Using the fact that $f,g$ are coprime and $K[z]$ is a PID then we may write $1 = f(z) p(z) + g(z) q(z)$ for some $p,q \in K[z]$ and from this I can express $c_1$ in terms that depend on the heights of $p,q$. What is the approach to relate the polynomial heights of $p,q$ to those of $f,g$ (and hence the height of $\phi$)?
The theory of resultants can be used to find a determinantal formula for $p$ and $q$ that involves the coefficients of $f$ and $g$. Then you can expand the determinant and use that to estimate the coefficients of $p$ and $q$.