Nagata's conjecture

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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

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

where ${\displaystyle \Delta =xz+y^{2}}$.

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