Nagata's conjecture
From HandWiki
Short description: Mathematical theorem in algebra
| Field | Algebraic geometry |
|---|---|
| Conjectured by | Masayoshi Nagata |
| Conjectured in | 1972 |
| First proof by | Ualbai Umirbaev and Ivan Shestakov |
| First proof in | 2004 |
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
- [math]\displaystyle{ \phi(x,y,z) = (x-2\Delta y-\Delta^2z,y+\Delta z,z), }[/math]
where [math]\displaystyle{ \Delta=xz+y^2 }[/math].
For the inverse, let [math]\displaystyle{ (a,b,c)=\phi(x,y,z) }[/math] Then [math]\displaystyle{ z=c }[/math] and [math]\displaystyle{ \Delta= b^2+ac }[/math]. With this [math]\displaystyle{ y=b-\Delta c }[/math] and [math]\displaystyle{ x=a+2\Delta y+\Delta^2 z }[/math].
References
- Nagata, Masayoshi (1972), On automorphism group of k[x,y], Tokyo: Kinokuniya Book-Store Co. Ltd., https://books.google.com/books?id=qvruAAAAMAAJ
- 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
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