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
FieldAlgebraic geometry
Conjectured byMasayoshi Nagata
Conjectured in1972
First proof byUalbai Umirbaev and Ivan Shestakov
First proof in2004

Nagata's automorphism is given by

where .

For the inverse, let Then and . With this and .

References

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  • 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