On the group of polynomial antomorphisms of the affine planes. (Sur le groupe des automorphismes polynomiaux du plan affine.)(English)Zbl 1096.14046

The main result of the paper under review is that, over a uncountable field $$k$$ of characteristic 0, the automorphism group of the group $$\text{Aut}(k^2)$$ of polynomial automorphisms of the affine plane is generated by the inner automorphisms of $$\text{Aut}(k^2)$$ and the action of automorphisms of the field: If $$\varphi$$ is an automorphism of the group $$\text{Aut}(k^2)$$, then there exists a polynomial automorphism $$\psi$$ and an automorphism $$\tau$$ of the field $$k$$ such that $$\varphi(f)=\tau(\psi f\psi^{-1})$$ for all $$f\in \text{Aut}(k^2)$$.
As an application, the author shows that an automorphism $$\varphi$$ of $$\text{Aut}(k^2)$$ is inner if and only if it preserves the determinant of the Jacobian matrix of the automorphisms of $$k^2$$. Another application gives that the automorphisms of the semigroup $$\text{End}(k^2)$$ of polynomial endomorphisms of $$k^2$$ are compositions of inner automorphisms and automorphisms of the field.

MSC:

 14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 13B25 Polynomials over commutative rings 14E07 Birational automorphisms, Cremona group and generalizations 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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References:

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