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Resumen de Classification of degree 2 polynomial automorphisms of C3

John Erik Fornaess Árbol académico, He Wu

  • For the family of degree at most $2$ polynomial self-maps of $\Bbb C^3$ 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 $abc\ne 0$.

    (iii) $$ \cases H_1(x,y,z)=(P(x,z)+ay,Q(z)+x,cz+d)\\ H_2(x,y,z)=(P(y,z)+ax,Q(y)+bz,y)\\ H_3(x,y,z)=(P(x,z)+ay,Q(x)+z,x)\\ H_4(x,y,z)=(P(x,y)+az,Q(y)+x,y)\\ H_5(x,y,z)=(P(x,y)+az,Q(x)+by,x) \endcases $$ where $P$ and $Q$ are polynomials with $\max\{\deg(P),\deg(Q)\}=2$ and $abc\ne 0$.

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