Nagata's conjecture

In algebra, Nagata's conjecture states that Nagata's automorphism of the polynomial ring k[x,y,z] is . 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 ${\displaystyle x=a+2\Delta y+\Delta ^{2}z}$.

References[]

• Nagata, Masayoshi (1972), On automorphism group of k[x,y], Tokyo: Kinokuniya Book-Store Co. Ltd., MR 0337962
• Umirbaev, Ualbai U.; Shestakov, Ivan P. (2004), "The tame and the wild automorphisms of polynomial rings in three variables", Journal of the American Mathematical Society, 17 (1): 197–227, doi:10.1090/S0894-0347-03-00440-5, ISSN 0894-0347, MR 2015334