Is there a differentiable surjective map from $\mathbb{Q}^m$ to $\mathbb{Q}^n$ where $n>m$?

Question

I want to know if there can exist a map of $n$ polynomials in $m$ variables, over the Rational numbers, $\mathbb{Q}^m$, such that if each polynomial corresponded to a coordinate in n dimensional Rational space, the map would be surjective.

If we were working over the Real numbers (or any smooth manifold), then we could use Sard's mini theorem to show that this type of map can't exist (because any such map would have measure $0$).

If we can show that a continuous differentiable map from $\mathbb{R}^m$ to $\mathbb{R}^n$, that maps all of the Rational values in $\mathbb{R}^m$ to the Rational values in $\mathbb{R}^n$ either

1. can not have measure zero,
2. must be closed,
3. must contain some ball of nonzero radius,

then my question will be answered.

Intuitively, I think that if such a map exists over the Rationals then it could be extended to a surjective map over the Reals, so I think that such a map can't exist, but I have been unable to prove it.

Any comments or ideas are appreciated.

Answer 1

There is such a smooth map. Let $S$ be any countable subset in $\mathbb{R}^n$, consisting of points $s_i$. To build a surjective smooth map from $\mathbb{Q}$ to $A$ just send $[2i, 2i+1]$ to $s_i$ and interpolate smoothly (using bump functions) on $[2i+1, 2i+2]$. Of course if you want a map from $\mathbb{Q}^m$ you can just project to $\mathbb{Q}$ first.

Polynomial maps is another kettle of fish entirely.

You can see that there is no such polynomial map from $\mathbb{Q}$ to any $\mathbb{Q}^n$ ($n>1$) by arguing along the lines outlined in your answer, because single-variable polynomial maps are proper. Any proper, continuous, surjective map $\mathbb{Q} \to \mathbb{Q}^n$ has a continuous, proper extension to a map $\mathbb{R} \to \mathbb{R}^n$ which will also be surjective (approximate $p\in \mathbb{R}^n$ by points $p_i$ in $\mathbb{Q}^n$; pick a sequence of their preimages; it will lie in some compact, and so subconverge to some $s$; that $s$ is sent to $p$).

In fact there is never a polynomial map of the kind you ask about for very general reasons: Surjective morphism of affine varieties and dimension, Surjective Morphism of affine algebraic variety etc.