On weighted bidegree of polynomial automorphisms of $\mathbb C^2$

Let $F=(F_1,F_2):\mathbb C^2\rightarrow \mathbb C^2$ be a polynomial automorphism. It is well known that $\deg F_1\,|\, \deg F_2$ or $\deg F_2\,|\, \deg F_1$. On the other hand, if $(d_1,d_2)\in \mathbb N_+^2=(\mathbb N\setminus \{ 0 \} )^2$ is such that $d_1\,|\, d_2$ or $d_2\,|\, d_1$, then one can construct a polynomial automorphism $F=(F_1,F_2)$ of $\mathbb C^2$ with $\deg F_1=d_1$ and $\deg F_2=d_2$.

Let us fix $w=(w_1,w_2)\in \mathbb N_+^2$ and consider the weighted degree on $\mathbb C[x,y]$ with $\deg_w x=w_1$ and $\deg_w y=w_2$. In this note we address the structure of the set $\{ (\deg_w F_1,\deg_w F_2) : (F_1,F_2)$ is an automorphism of $\mathbb C^2\}$. This is a very first, but necessary, step in studying weighted multidegrees of polynomial automorphisms.