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I am searching the polynomial functions from $S^3$ to $S^3$.

We say $g$ is a polynomial function from $S^3$ to $S^3$, if there exists $f_1,f_2,f_3,f_4 \in \mathbb{R}[X_1,X_2,X_3,X_4]$ and $f:\mathbb{R}^4 \rightarrow \mathbb{R}^4$, $(x_1,x_2,x_3,x_4) \mapsto (f_1(x_1,x_2,x_3,x_4),...,f_4(x_1,x_2,x_3,x_4))$ such that $g: S^3 \rightarrow S^3$, maps $(x_1,x_2,x_3,x_4)$ to $ f(x_1,x_2,x_3,x_4)$

We generate the set $F$ of functions from $S^3$ to $S^3$ by:

  • the constant functions belong to $F$,

  • the identity belongs to $F$,

  • the isometries belong to $F$,

  • if $f,g \in F$, $\overline{f} \in F$, and $f \circ g\in F$, and $ f \times g\in F$, where $\overline{z}$ is the conjuguate of $z$ in the quaternions, and $\times$ is the quaternions product.

Are the functions of $F$ all the polynomial functions from $S^3$ to $S^3$ ?

Thanks in advance.

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Any advances in this problem? BTW, construction rules are overly redundant: the identity map is an “isometry”, and quaternion conjugation is also an ℝ-linear map expressed with a diagonal matrix that obviously belongs to O(4). –  Incnis Mrsi Nov 9 at 15:12

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