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