title: Nagata's conjecture
text: # 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 $x=a+2\Delta y+\Delta ^{2}z$.
  
  