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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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