I have this question after reading Andrew Bridy's recent paper, Automatic Sequences and Curves Over Finite Fields. I would like to understand the following relationships used in the proof of Corollary 3.10:
$$ \deg \pi_x = \deg (x)_\infty \text { and } \deg \pi_y = \deg (y)_\infty. \tag{$*$}$$
Background:
Let $k$ be a perfect field of positive characteristic. Let $y \in k((x))$ be a Laurent series which is algebraic over $k(x)$. Let $X$ be the normalization of the projective closure of the curve defined by the minimal polynomial of $y$. Let $\pi_x, \pi_y : X \to \mathbf{P}^1$ be the projections of $X$ onto the $x$ and $y$ coordinates respectively. (These are dual to the inclusions of $k(x)$ and $k(y)$ into $k(X)$.) For $f \in k(X)$, the symbol $(f)_\infty$ denotes the divisor corresponding to the poles of $f$, i.e.
$$ (f)_\infty = \sum_{\substack{P \in X \\ v_P(f) < 0}} -v_{P}(f) \cdot P. $$
Bridy claims the identity $(*)$ in Corollary 3.10.
I know that $\deg (f)_\infty$ is just the sum of the orders of the poles of $f$ and I know that if $g(x)$ is a polynomial then $g$ has a pole of order $\deg g$ at $\infty$, which I suspect is related. What I would like is for a more precise explanation. I know basic properties of schemes, projective varieties and divisors but I'm far from an expert.