A question about the existence of rational functions

I am reading a paper Representations of shifted quantum affine algebras. I have a question about the existence of a rational function about the remark $4.4$

I'll briefly describe the problem.

We let $\phi^+(z)=\sum_{m\ge 0}\phi_m z^m$ and $\phi^-(z)=\sum_{m\ge -q}\phi^-_{-m}z^{-m}$,where $q\in \mathbb{Z}$.
In the paper the author proves that there exists a polynomial $P(z)$ satisfies $P(z)(\phi^+(z)-\phi^+(z))=0$.
Then the author says that $\phi^+(z)$ and $\phi^-(z)$ can be seen as one rational function expanding at $z$ and $z^{-1}$ respectively.

I think this means there exists a rational function $f(z)\in \mathbb{C}(u)$ such that $f(u)=\phi^+(z)=\phi^-(z)$ from $P(z)(\phi^+(z)-\phi^-(z))=0$

My question :

How can I get there exists a rational function $f(z)\in \mathbb{C}(u)$ such that $f(u)=\phi^+(z)=\phi^-(z)$ from $P(z)(\phi^+(z)-\phi^-(z))=0$?

Any help and references are greatly appreciated.

Thanks!