Nagata's conjecture

For the conjecture about curves, see Nagata's conjecture on curves.

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 conjecture

Field	Algebraic geometry
Conjectured by	Masayoshi Nagata
Conjectured in	1972
First proof by	Ualbai Umirbaev and Ivan Shestakov
First proof in	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