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On polynomial automorphisms of spheres. (English) Zbl 1194.14083

The authors study \(\operatorname{Aut}(S^n(K))\), the automorphism group of the \(n\)-sphere over \(K\), an infinite field of characteristic different from two. In particular, they study the relation of \(\operatorname{Aut}(S^n(K))\) and \(\operatorname{Aut}(K^{n+1},S^n(K))\), the latter being the automorphisms of \(K^n\) preserving \(S^n(K)\).
Some results of the article:
– If \(K\) is infinite and not of characteristic 2, then \(\operatorname{Aut}(K^{n+1},S^n(K))\) equals \[ \{\varphi \in \operatorname{Aut}(K^{n+1})~|~\varphi(x_0^2+\ldots+x_n^2)=x_0^2+\ldots+x_n^2\}. \]
– If \(K\) is a real field (that is, \(x_1^2+\ldots+x_n^2=-1\) has no solutions) then \(\operatorname{Aut}(K^{n+1},S^n(K))=O(n+1,K)\), the set of orthogonal linear maps.
– \(\operatorname{Aut}(K^2,S^1(K))\) is isomorphic to \(\operatorname{Aut}(S^1(K))\).
– \(\operatorname{Aut}(K^3,S^2(K))\longrightarrow \operatorname{Aut}(S^2(K))\) is a surjection for any algebraically closed field \(K\) with characteristic different from two.

MSC:

14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
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References:

[1] M. Golasiński and F. Gómez Ruiz, On maps of tori, Bulletin of the Belgian Mathematical Society – Simon Stevin 13 (2006), 139–148. · Zbl 1159.14035
[2] M. W. Hirsch, Automorphisms of compact affine varieties, in Global Analysis in Modern Mathematics (Orono, ME, 1991; Walthom, MA, 1992), Publish or Perish, Houston, TX, 1993, pp. 227–245. · Zbl 0929.57025
[3] Z. Jelonek, Identity sets for polynomial automorphisms, Journal of Pure and Applied Algebra 76 (1991), 333–337. · Zbl 0752.14010 · doi:10.1016/0022-4049(91)90141-N
[4] H. Jung, Über ganze birationale Transformationen der Ebene, Journal für die Reine und Angewandte Mathematik 184 (1942), 161–174. · Zbl 0027.08503 · doi:10.1515/crll.1942.184.161
[5] W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde 3 (1953), 33–41. · Zbl 0050.26002
[6] L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel Journal of Mathematics 69 (1990), 250–256. · Zbl 0711.14022 · doi:10.1007/BF02937308
[7] L. Makar-Limanov, On the group of automorphisms of a surface x n y = P(z), Israel Journal of Mathematics 121 (2001), 113–123. · Zbl 0980.14030 · doi:10.1007/BF02802499
[8] H. Matsamura and P. Monsky, On automorphisms of hypersurfaces, Journal of Mathematics of Kyoto University 3 (1964), 347–361. · Zbl 0141.37401
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