Connexion between the number of poles of a function and the degree of the associated projection map

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.

• Not a full answer, but: this basically follows from the "fundamental identity" in algebraic number theory. Let $\mathfrak{p}$ be a place of $k(x)$ ($\mathfrak{p} = \infty$ in your case), let $\{\mathfrak{P}_1, \ldots, \mathfrak{P}_g\}$ be the set of places of $k(X)$ lying above $\mathfrak{p}$, and let $e_i$ and $f_i$ be the associated ramification indices and residue degrees. Then $\sum_{i=1}^g e_i f_i = [k(X): k(x)]$ (and by definition $\deg(\pi_x) = [k(X): k(x)]$). This is proved as Theorem 7.6 in Rosen's Number Theory in Function Fields. – André 3000 Jan 25 '18 at 4:21
• @Quasicoherent That helps a lot. Thank you! – Trevor Gunn Jan 25 '18 at 5:04