On polynomial automorphisms of spheres

Abstract

Let K be an infinite field with characteristic different from two and $\mathbb{S}^n$ (K) the n-sphere over K. We show that ambient polynomial automorphisms of $\mathbb{S}^n$ (K) preserve the quadratic form x ²₀ + ⋯ + x ² _n and the group Aut ((K ⁿ⁺¹, $\mathbb{S}^n$ (K)) of such automorphisms of $\mathbb{S}^n$ (K) is isomorphic to the (n + 1)-orthogonal group O(n + 1, K) provided K is real.

Next, the restriction map Aut (K ³, $\mathbb{S}^2$ (K)) → Aut ($\mathbb{S}^2$ (K)) yields a surjection provided K is an algebraically closed field as well. Furthermore, for any such a field K, there is an imbedding

$$ K[X_1 \ldots ,X_m ]^{\tfrac{{m(m - 1)}} {2} + mn} \to Aut (K^{2m + n} ,\mathbb{S}^{2m + n - 1} (K)) $$

.