For the family of degree at most 2 polynomial self-maps of C3 with nowhere vanishing Jacobian determinant, we give the following classification: for any such map f, it is affinely conjugate to one of the following maps:
(i) An affine automorphism;
(ii) An elementary polynomial autormorphism
E(x, y, z) = (P(y, z) + ax, Q(z) + by, cz + d),
where P and Q are polynomials with max{deg(P), deg(Q)} = 2 and abc ≠ 0.
(iii)
⎧ H1(x, y, z) = (P(x, z) + ay, Q(z) + x, cz + d)
⎪ H2(x, y, z) = (P(y, z) + ax, Q(y) + bz, y)
⎨ H3(x, y, z) = (P(x, z) + ay, Q(x) + z, x)
⎪ H4(x, y, z) = (P(x, y) + az, Q(y) + x, y)
⎩ H5(x, y, z) = (P(x, y) + az, Q(x) + by, x)
where P and Q are polynomials with max{deg(P), deg(Q)} = 2 and abc ≠ 0.
@article{urn:eudml:doc:41327,
title = {Classification of degree 2 polynomial automorphisms of C3.},
journal = {Publicacions Matem\`atiques},
volume = {42},
year = {1998},
pages = {195-210},
mrnumber = {MR1628170},
zbl = {0923.58006},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41327}
}
Fornaess, John Erik; Wu, He. Classification of degree 2 polynomial automorphisms of C3.. Publicacions Matemàtiques, Tome 42 (1998) pp. 195-210. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41327/