Let $F,G,H,K \in \mathbb{C}[x,y]-\mathbb{C}$.
Assume that the following fields of fractions are equal: $\mathbb{C}(F,G)=\mathbb{C}(H,K)$.
Claim I: There exist $a,b,c,d \in \mathbb{C}$ such that the following ideals are equal: $\langle F-a,G-b \rangle=\langle H-c,K-d \rangle$.
Question I: Is the above Claim I true?
I have checked a few examples, but I do not know how to prove the claim. Perhaps there is a counterexample?
Examples for a positive answer:
(1) $\mathbb{C}(x,y)=\mathbb{C}(x,y+x^3+7x^2+5x-1)$ and $\langle x,y \rangle=\langle x,y+x^3+7x^2+5x-1+1 \rangle$.
Here $a=b=c=0, d=-1$.
(2) $\mathbb{C}(x,y)=\mathbb{C}(x+(x-y)^2,y+(x-y)^2)$ and $\langle x,y \rangle=\langle x+(x-y)^2,y+(x-y)^2 \rangle$.
Here $a=b=c=d=0$.
This question is relevant; the current question is a generalization of the second statement in the quoted question, changing $x,y$ to arbitrary pair of non-constant polynomials.
The following is a generalization of the current question:
Let $F_1,\ldots,F_n,G_1,\dots,G_m \in \mathbb{C}[x,y]-\mathbb{C}$.
Assume that the following fields of fractions are equal: $\mathbb{C}(F_1,\ldots,F_n)=\mathbb{C}(G_1,\ldots,G_m)$.
Claim II: There exist $a_1,\ldots,a_n,b_1,\ldots,b_m \in \mathbb{C}$ such that the following ideals are equal: $\langle F_1-a_1,\ldots,F_n-a_n \rangle=\langle G_1-b_1,\ldots,G_m-b_m \rangle$.
Question II: Is the above Claim II true?
Thank you very much!