In André Weil's dissertation, he considers two meromorphic functions $x,y$ on a complex curve.  He assumes every pole of $y$ is a pole of $x$, and its multiplicity as a pole of $y$  is no greater than its multiplicity as a pole of $x$.  Then he says there is some natural number $k$ and some complex $a\neq 0$ such that 
$$ay^k+xP(x,y)+Q(y)=0$$
where $P(x,y)$ and $Q(y)$ are polynomials in $x,y$ with degree $<k$.  


I see the proof for genus 0 curves. There the field of meromorphic functions is $\mathbb{C}(z)$ and one-variable polynomial algebra suffices (unless I've made a mistake). But I do not see it for other curves. Can someone tell me how it is done?



Some simple algebra of rational functions by André Weil
Let us write the irreducible equation relating $x$ and $y$ as
$$P\_k(x)y^k+P\_{k-1}(x)y^{k-1}+\ldots+P\_0(x)=0.$$
Consider the Newton polygon (the graph of the smallest concave function $\phi$ with
$\phi(j)\geq \deg P\_j,\; 0\leq j\leq k$. 
Condition on the poles of $x$ and $y$ tells us that 
$P\_k=\mathrm{const}$, and all
slopes of this graph are $\geq -1.$ This implies that
$$\deg P\_j\leq m-j,\quad 0\leq j\leq k-1.$$ 
Now unite all constant terms of $P\_j$ times powers of $y$ into
the polynomial $Q(y)$, $\deg Q<k$, and the rest is $xP(x,y)$, where degree of
$P$ with respect to $x$ and $y$ is at most $k-1$.


