Let $K$ be an algebraically closed field (or $\overline{\mathbb{C}(z)}$ for more requirement). And let $P \in K[x,y]$ be an irreducible polynomial of degree $m$ with respect to $x$ and degree $n$ with respect to $y$. Consider the function field $R:=Frac(K[x,y]/(P))$ associated to $P$.

I would like to know why $\deg \delta^{-} (x) = n$ and $\deg \delta^{-} (y) = m$,

where for $t \in R$, $\delta (t)$ is denoted by the sum $\sum (ord_p(t)).p$ with p runs all over places of $R$; and $\delta^- (t)$ the negative part only of $\delta (t)$; and $\deg \delta(t):= \sum ord_p(t)$. Any help would be greatly appreciated!

Let $K$ be an algebraically closed field (or $\overline{\mathbb{C}(z)}$ for a more precise condition). And let $P \in K[x,y]$ be an irreducible polynomial of degree $m$ with respect to $x$ and degree $n$ with respect to $y$. Consider the function field $R:=Frac(K[x,y]/(P))$ associated to $P$.

I would like to know why $\deg \delta^{-} (x) = n$ and $\deg \delta^{-} (y) = m$. Here, for $t \in R$, $\delta (t)$ denotes the sum $\sum (ord_p(t)).p$ with p running over all places of $R$; and $\delta^- (t)$ is the negative part only of $\delta (t)$; and $\deg \delta(t):= \sum ord_p(t)$. Any help would be greatly appreciated!