Nagata's conjecture

In algebra, Nagata's conjecture states that Nagata's automorphism of the polynomial ring k[x,y,z] is wild. The conjecture was proposed by and proved by.

Nagata's automorphism is given by
 * $$\phi(x,y,z) = (x-2\Delta y-\Delta^2z,y+\Delta z,z),$$

where $$\Delta=xz+y^2$$.

For the inverse, let $$(a,b,c)=\phi(x,y,z)$$ Then $$z=c$$ and $$\Delta= b^2+ac$$. With this $$y=b-\Delta c$$ and $$x=a+2\Delta y+\Delta^2 z$$.