Let $k$ be a field of characteristic zero (for example, $k= \mathbb{C}$), and let $f(x) \in k[x]$ and $g(y) \in k[y]$ with $\deg_x(f(x)) \geq 1$ and $\deg_y(g(y)) \geq 1$.
Let $p,q \in k[x,y]$, and denote by $I$ the ideal generated by $\{p,q\}$, $I:=\langle p,q \rangle$.
Further assume that $\{p,q\}$ are of the following forms: $p= f_1+g_1+xy H_1$ and $q= f_2+g_2 +xy H_2$, where $f_1,f_2 \in k[x]$, $g_1,g_2 \in k[y]$, $H_1,H_2 \in k[x,y]$ with $\langle f_1,f_2 \rangle =1$ and $\langle g_1,g_2 \rangle =1$.
If it happens that $f(x),g(y) \in I$, does this imply something interesting about $p$ and $q$?
In other words: $f(x) \in I$ means that there exist $A,B \in k[x,y]$ such that $f(x)=Ap+Bq$ and $g(y) \in I$ means that there exist $C,D \in k[x,y]$ such that $g(y)=Cp+Dq$.
A somewhat relevant question is this, but notice that here I do not assume that $I$ is a prime ideal (and $\{f(x),g(y)\}$ are not relatively prime).
Any hints and comments are welcome! Thank you.