Let $C$ be the plane curve defined by $y^d =f(x)$, where $f(x) \in \mathbb{C}[x]$ is a polynomial. Let $g(x,y)$ and $h(x,y)$ are polynomials relatively prime with each other. We consider $r(x,y) =g(x,y)/h(x,y)$ as a function on $C$, and we assume that $r(x,y)$ is not constant on $C$. If for infinitely many $y \in \mathbb{C}$, there exist at least two points $(x_1,y)$ and $(x_2,y)$ on $C$ such that $r(x_1,y)=r(x_2,y)$, then can we claim that there exist polynomials $g'(y)$ and $h'(x)$ in $\mathbb{C}[y]$ such that $r(x,y) =g'(y)/h'(y)$ on $C$?
Thanks!
Thanks to the help of Ferra, I have known this is not true. But I am still wondering whether there is some characteristic on the expressions of the functions $r(x,y)$ satisfying the property above?
No. Take for example $y^d=x^4-1$ and $r(x,y)=y+x^2$. Then a point $(x,y)\in C$ iff $(-x,y)\in C$ and so for all $y\in \mathbb C$ you have two points $(x,y),(-x,y)\in C$ which are such that $r(x,y)=r(-x,y)$. But $r$ cannot be a function only of $y$, since for example $(1,0)$ and $(i,0)$ are both points of $C$ with the same $y$ but $r(1,0)=1\neq -1=r(i,0)$.