Nagata's conjecture

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.