Let $f,g \in \mathbb{C}[x,y]$. A comment to this question says that it is possible to write $g=qf+r$, for some $q,r \in k[x,y]$, but not necessarily with $\deg(r) < \deg(f)$.
Is there something interesting that can be said about $r$?
Moreover, its first answer says: "But all Euclidean is not lost, since one can generalize the polynomial division algorithm in a way that recovers many of the important properties".
Which properties?
In particular, we know that if $\gcd(f,g)=1$, then it is not true that necessarily there exist $u,v \in k[x,y]$ such that $uf+vg=1$, as for example $f=x$, $g=y$ show.
Thank you very much!