Let $k[x,y]$ be bivariate polynomial ring and $k$ be a field. Let $A,B,C,D \in k$ be scalars and $f,g \in k[x,y]$ be two polynomials. Assume that the following equations hold:
- $B^2 f = D^2 g$
- $ABf = CDg$
What is the relation between the scalars $A,B,C,D$?
Here is what I did: From the first equation, we have $f = D^2 h$ and $g = B^2 h$ for some polynomial $h \in k[x,y]$. Writing these in the second equation gives $ABD^2h = CDB^2h$ and hence, the relation is $AD = BC$. Does it look correct to you? I know this might be a silly question and sorry for that. Thanks!