Nagata's conjecture explained

Nagata's conjecture
Field:	Algebraic geometry
Conjectured By:	Masayoshi Nagata
Conjecture Date:	1972
First Proof By:	Ualbai Umirbaev and Ivan Shestakov
First Proof Date:	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 and proved by .

Nagata's automorphism is given by

\phi(x,y,z)=(x-2\Deltay-\Delta^2z,y+\Deltaz,z),

where

\Delta=xz+y²

For the inverse, let

(a,b,c)=\phi(x,y,z)

Then

z=c

and

\Delta=b^2+ac

.With this

y=b-\Deltac

and

x=a+2\Deltay+\Delta²z