Let $k$ be a field of characteristic zero. Let $a,b,c,d \in k[x,y]$ be four polynomials, each two are algebraically independent over $k$. Let $R:=k[a+b,c+d]$ be of transcendence degree two over $k$ (so $a+b,c+d,b$ are algebraically independent over $k$).
Since the transcendence degree of $k[x,y]$ is two, every three elements of $k[x,y]$ are algebraically dependent over $k$, in particular, $b,a+b,c+d$ are algebraically dependent over $k$, so $b$ is algebraic over $k[a+b,c+d]$.
Is it possible to somehow characterize the minimal polynomial of $b$ over $k[a+b,c+d]$?
Example: $a=x+x^2+y^2$, $b=-2xy$, $c=x^2+y^2$, $d=y-2xy$. Then $a+b=x+x^2+y^2-2xy=x+(x-y)^2$ and $c+d=x^2+y^2+y-2xy=y+(x-y)^2$.
What is the minimal polynomial $m_{-2xy}(T) \in k[a+b,c+d][T]$ of $-2xy$ over $k[a+b,c+d]=k[x+(x-y)^2,y+(x-y)^2]=k[x,y]$? It is of course $m_{-2xy}(T)=T+2xy \in k[x,y][T]$.
Any hints and comments are welcome!