Does the $n$-sphere have a *surjective* rational parameterization $p \colon \mathbb{R}^k \to S^n$ for some $k \geq 1$? For each $n \geq 1$, does the $n$-sphere
$$S^n = \{x \in \mathbb{R}^{n+1} : x^\top x = 1\}$$
have a surjective rational parameterization?  That is, does there exist some $k \geq 1$ and a surjective rational map $p \colon \mathbb{R}^k \to S^n$?
Thoughts:

*

*I believe it is crucial that I have specified the domain to be $\mathbb{R}^k$ (i.e., contractible), because otherwise you could just cover $S^n$ in patches and then take a parameterization which is surjective on each patch.

*My guess is that the answer is no for each $n \geq 1$, but I'm not sure how to prove it, even in the case $n=1$.

 A: $\newcommand{\Numbers}[1]{\mathbf{#1}}\newcommand{\Cpx}{\Numbers{C}}\newcommand{\Reals}{\Numbers{R}}$
tl; dr: Yes, there exist rational surjections $\Reals^{n} \to S^{n}$ for every positive integer $n$. (The construction below gives immersions/submersions if and only if $n = 1$. There also exist surjective rational submersions $\Reals^{2m+2} \to S^{2m+1}$, and "exceptional" submersions $\Reals^{2n} \to S^{n}$ for $n = 2$, $4$, and $8$.)

As is surely known to OP and commenters, inverse stereographic projection is a rational diffeomorphism from a Cartesian space to a punctured sphere of the same dimension. A natural approach is therefore to seek a rational map from the sphere to itself that "covers the missing point."
For $n = 1$ such mappings exist, because the complex power functions are polynomial and map the unit circle to itself. Thus, for example, the composition
\begin{align*}
  t &\mapsto (x, y) := \frac{(2t, t^{2} - 1)}{t^{2} + 1} \\
  &\mapsto (x^{2} - y^{2}, 2xy)
  = \frac{(-t^{4} + 6t^{2} - 1, 4t(t^{2} - 1))}{(t^{2} + 1)^{2}}
\end{align*}
of inverse stereographic projection and squaring is a rational surjection from the line $\Reals$ to the unit circle $S^{1}$, among infinitely many others.
By taking Cartesian products, we get obvious rational surjections $\Reals^{n} \to (S^{1})^{n}$ of the real $n$-torus sitting in $\Reals^{2n}$. It suffices to show there is a polynomial mapping whose restriction to the $n$-torus is the unit $n$-sphere. For concreteness, let's consider the case $n = 2$, and write the $2$-torus as
$$
\{(u, v, x, y) : u^{2} + v^{2} = 1 = x^{2} + y^{2}\}.
$$
The polynomial mapping $(u, v, x, y) \mapsto (xu, xv, y)$ effects spherical coordinates, and is surjective because both $(u, v)$ and $(x, y)$ range over the whole unit circle. The usual iterative description of spherical coordinates extends this construction to arbitrary dimension.

There are noteworthy alternative rational surjections $\Reals^{k} \to S^{n}$ if

*

*$n = 2m + 1$ is odd and $k = n + 1 = 2(m + 1)$;

*$k = 2n$ and $n = 2$, $4$, or $8$.

Taking $n = 2m + 1$ and $k = n + 1$, we can use $\Reals^{n} = \Reals^{2m+1}$ as stereographic parametrization of the punctured sphere inside $\Reals^{2m+2} \simeq \Cpx^{m+1}$, use $\Reals$ as surjective parametrization of $S^{1}$ as above, and finally use the (polynomial) action of complex multiplication $\Cpx \times \Cpx^{m+1} \to \Cpx^{m+1}$ to get a surjection to the sphere $S^{n} = S^{2m+1}$.
Finally, there are three exceptional instances where the preceding ideas work: A rational surjection $\Reals^{4} \to S^{3}$ gives rise to a rational surjection to $S^{2}$ by the Hopf map. Similarly, the quaternionic Hopf map $S^{7} \to S^{4}$ gives a rational surjection $\Reals^{8} \to S^{4}$, and the octonionic Hopf map a rational surjection $\Reals^{16} \to S^{8}$.
