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 Template:Harvs and proved by Template:Harvs.

Nagata's automorphism is given by

$\varphi (x,y,z)=(x-2\mathrm{\Delta }y-{\mathrm{\Delta }}^{2}z,y+\mathrm{\Delta }z,z),$

where $\mathrm{\Delta }=xz+y^2$.

For the inverse, let $(a,b,c)=\varphi (x,y,z)$ Then $z=c$ and $\mathrm{\Delta }=b^2+ac$. With this $y=b-\mathrm{\Delta }c$ and $x=a+2\mathrm{\Delta }y+{\mathrm{\Delta }}^{2}z$.

References

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