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 automorphism is given by

$(x,y,z)\mapsto (x-2(xz+y^{2})y-(xz+y^{2})^{2}z,y+(xz+y^{2})z,z).$

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