Consider the function $G:\mathbb{Q}^{n} \rightarrow \mathsf{FinGroup}$ which sends $\langle a_0,a_1,\dots,a_{n-1}\rangle$ via the polynomial $P(x) = a_{n-1}x^{n-1} + \dots + a_1x + a_0$ to its Galois group.
How does this function partition $\mathbb{Q}^{n}$?
Edit: A little more thinking led me to the conclusion that it might be more appropriate to consider functions $G:\mathbb{Q}\mathbb{P}^{n} \rightarrow \mathsf{FinGroup}$ with $\mathbb{Q}\mathbb{P}^{n}$ something like the rational projective space with points = straight lines in $\mathbb{Q}^{n}$ going through the origin, due to the fact that $G(\vec{a}) = G(c\vec{a})$ for each $c \in \mathbb{Q}\setminus \{0\}$.
Does the following still make sense then (replacing $\mathbb{Q}^n$ by $\mathbb{Q}\mathbb{P}^{n}$)? Can we do epsilontics in projective spaces?
There may be $\vec{a} \in \mathbb{Q}^{n}$ and $\epsilon > 0$ with $G(\vec{a}) = G(\vec{b})$ for all $\vec{b}$ with $|\vec{a}-\vec{b}| < \epsilon$, and there may be $\vec{a}$ for which this doesn't hold. There even may be isolated points $\vec{a}$ such that for no $\epsilon > 0$ there is a $\vec{b}$ with $|\vec{a}-\vec{b}| < \epsilon$ and $G(\vec{a}) = G(\vec{b})$.
Can I somehow imagine this partition, resp. the shapes of the borders between regions with constant $G$? The first non-trivial case is $n = 4$ for cubic polynomials having three possible Galois groups: the trivial group, $S_3$ and $A_3$. What can be said about the "distribution" of these three groups over $\mathbb{Q}^{4}$?