Nagata's conjecture
# 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 Nagata (1972) and proved by Ualbai U. Umirbaev and Ivan P. Shestakov (2004).

Nagata's automorphism is given by

    $\phi (x,y,z)=(x-2\Delta y-\Delta ^{2}z,y+\Delta z,z),$

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 $x=a+2\Delta y+\Delta ^{2}z$.

