Which rational functions are regular at the point of infinity of $ \mathbb{P}^{1}?$ Which order of zero do they have there?
This is an exercise in Shafarevich's "Basic Algebraic Geometry I".
This question has been partially answered here Which rational functions $\mathbb{P}^1\rightarrow k$ are regular at the point at infinity? and here: Regularity of rational functions at infinity
Such a rational function is some $ F(X,Y) = \frac{G(X,Y)}{H(X,Y)} $ where $ G $ and $ H $ are homogeneous equations of equal degree, and $ H(X,Y) \neq 0. $ Can more be said about this? I don't see how we can be more specific.
My understanding is that the order of the zero is the integer $ \text{deg}(H(X,Y))-\text{deg}(G(X,Y)). $
Is this not always equal to zero? What am I missing?
The only rational functions $p(x)/q(x)$ which are regular at the point at infinity of $\mathbb{P}^1$ are the zero rational function, and those non-zero rational functions for which $\deg q \geq \deg p$. Indeed, it is clear that the zero rational funtion is regular everywhere on $\mathbb{P}^1$, including the point at infinity. Now consider any non-zero rational function $p(x)/q(x)$ (in inhomogeneous coordinates). Suppose $p(x) = \sum_{i = 0}^n a_ix^i$ and $q(x) = \sum_{i = 0}^m b_i x^i$. Then we can express the rational function $p(x)/q(x)$ in terms of the homogeneous coordinates $(X:Y)$ as
\begin{equation} \frac{u(X,Y)}{v(X,Y)} := \frac{\sum_{i = 0}^n a_iX^iY^{M-i}}{\sum_{i = 0}^m b_iX^iY^{M-i}}, \end{equation}
where $M = \max(n,m)$. Indeed, note that, as defined above, $u$ and $v$ are homogeneous polynomials is $k[X,Y]$ both of degree $M$. Moreover, if we consider a point $(x:1)$ other than the point at infinity at which $q(x) \neq 0$,
\begin{equation} \frac{u(x,1)}{v(x,1)} = \frac{\sum_{i = 0}^n a_ix^i}{\sum_{i = 0}^m b_ix^i} = \frac{p(x)}{q(x)}. \end{equation}
The rational function $\frac{u(X,Y)}{v(X,Y)}$ is regular at the point at infinity $(1:0)$ if and only if $v(1,0) \neq 0$, which is true if and only if $M = m \iff m \geq n$. The order of zero at the point at infinity is given by $\deg q - \deg p = m - n$.