Let $u,v \in \mathbb{C}[x]$.
Assume that $\mathbb{C}(u,v)=\mathbb{C}(x)$. Then by this answer, there exists $\alpha,\beta,\gamma \in \mathbb{C}$ such that $\langle u-\alpha,v-\beta \rangle = \langle x-\gamma \rangle$, where $\langle * \rangle$ is the ideal generated by $*$.
Is this result can be generalized to $\mathbb{C}[x,y]$?
First attempt Let $s,t,u,v \in \mathbb{C}[x,y]$. Assume that $\mathbb{C}(s,t,u,v)=\mathbb{C}(x,y)$. Is it true that there exist $\alpha,\beta,\gamma,\delta,\epsilon_1,\epsilon_2 \in \mathbb{C}$ such that $\langle s-\alpha,t-\beta,u-\gamma,v-\delta \rangle = \langle x-\epsilon_1,y-\epsilon_2 \rangle$?
Second attempt: Let $u,v \in \mathbb{C}[x,y]$. Assume that $\mathbb{C}(u,v)=\mathbb{C}(x,y)$. Is it true that there exist $\alpha,\beta,\gamma,\delta \in \mathbb{C}$ such that $\langle u-\alpha,v-\beta \rangle = \langle x-\gamma,y-\delta \rangle$?
Are there counterexamples for both attempts?
Thank you very much!
Edit: What happens if we replace $y$ by $y^2$, namely: Let $u,v \in \mathbb{C}[x,y]$. Assume that $\mathbb{C}(u,v)=\mathbb{C}(x,y^2)$. Is it true that there exist $\alpha,\beta,\gamma,\delta \in \mathbb{C}$ such that $\langle u-\alpha,v-\beta \rangle = \langle x-\gamma,y^2-\delta \rangle$? It seems that this result is also true or am I missing something?
Let $A\subset B$ be integral domains with $B$ a finitely generated $A$- algebra. Assume that the fraction fields are the same. Then one can find an $s\neq 0$ in $A$ such that $A_s=B_s$. Now you can easily check that for any maximal ideal $M$ of $A$ not containing $s$, $MB$ is a maximal ideal of $B$. In your case, further use Hilbert Nullstellensatz to get what you want.