Let $f$ be a "typical" smooth non-polynomial map from $\mathbb{R}^3$ to $\mathbb{R}^7$. Is it reasonable to expect that $f(\mathbb{R}^3)$ is not included within the zero-set of a system of polynomial equations, i.e. not included in an proper algebraic subset of $\mathbb{R}^7$?
Is there any explicit criterion/sufficient condition on $f$ to make sure this does not happen?
Or conversely, if there is a system of polynomial $Q$ such that $Q(f)=0$, does it tell us anything particular about $f$?
Edit: of course I don't care about 3 and 7 in particular.
Consider just the case $\mathbb{R}^1 \to \mathbb{R}^n$, i.e. smooth curves in $\mathbb{R}^n$. It's not hard to come up with a small arc whose Zariski closure is all of $\mathbb{R}^n$. By replacing an arbitrarily small arc of any given curve with this bad arc, we get a curve which cannot be contained in any proper algebraic subset of $\mathbb{R}^n$. So at a minimum we see that the curves which cannot fit in a proper algebraic subset are dense among all curves.
Moreover, if you have a large family of curves, each of which fits in some real algebraic hypersurface, you can do this to all of them simultaneously and get an equally large family of curves, none of which fit in any real algebraic hypersurface. In fact, you can do it in lots of different ways. I'm fairly sure that rules out the former class having a positive measure in any meaningful sense, but I'm not a functional analyst.