Minimum dimension of manifold domains of functions that generate all the roots of polynomials of degree equal (or at most) $k$

Question

EDIT: I have realized that there were some problems in some of the definitions, which were not quite what I meant, so I fixed them.

I have a question that I have been wondering since always, but I am not aware of any literature about it. We know by Galois theory that it is not possible, in general, to solve for the roots of polynomials by extracting radicals. If we allow for any arbitrary function of the coefficients of the polynomials, I was wondering, given a fixed positive integer $k$, what is the minimum dimension of a manifold (or union of manifolds) that are the domain(s) of some generating functions,$h_1,\dots,h_l$, such that, starting from $h_1,\dots,h_l$ and applying recursively algebraic, rational operations as well as compositions, one can eventually find a finite set of functions $f_1,\dots,f_h$ that generate all the roots of all the polynomials of degree equal (or not greater) than $k$, in the sense that I clarify formally below.

First of all, I make some preliminary definitions:

• $X := \bigcup_{m, n \in \mathbb{N}}X_m^n$, $X^n := \bigcup_{m \in \mathbb{N}}X_m^n$, where $X_m^n:=\{f:A \rightarrow \mathbb{C}^n, A \subseteq \mathbb{C}^m \}$;
• $\forall n \in \mathbb{N}, C^{n}[x]:=\{p \in C[x], \mbox{deg}(p) = n\}, C^{\leq n}[x]:=\{p \in C[x], \mbox{deg}(p) \leq n\}$
• $\pi:\mathbb{C}[x] \rightarrow \bigcup_{n\in \mathbb{N}}\mathbb{C}^n$, $\pi_i:\mathbb{C}[x] \rightarrow \mathbb{C}$, $\forall p \in \mathbb{C}[x]$, $p(x)=a_0 + a1 \cdot x + ... + a_k \cdot x^k$, $\pi(p) := (a_0, a_1,...,a_k)$, $\pi_i(p):=a_i, \forall i \in \{1,\dots,k\}$;
• $\forall i,k \in \mathbb{N}, i \leq k-1, \tilde{\pi}_{i,k}:\mathbb{C}^k \rightarrow \mathbb{C}, \tilde{\pi}_{i,k}(a_0,\dots,a_{k-1}):=a_i$
• $\forall m,n \in \mathbb{N}^+$, $\mathcal{i}_{m,n}:\left(\mathbb{C}^m \times \mathcal{C}^n \right) \rightarrow \mathbb{C}^{m+n}$, $\mathcal{i}_{m,n}((a_0,\dots,a_{m-1}),(b_0,\dots,b_{n-1})) = (a_0,\dots,a_{m-1},b_0,\dots, b_{n-1})$;
• for any finite set $A$, $vec(A) = (x_1, \dots, x_k)$ if $A = \{x_1, \dots, x_k\}$, $\forall i,j \in \{1,\dots,k\}$, if $x_i = x_j \Rightarrow i=j$.

Given $\mathcal{H} \subseteq \mathcal{A} \subseteq X$, we say that $\mathcal{H}$ is an Algebrically and Compositionally Closed Set of Functions in $\mathcal{A}$ - in short $\mathcal{H}$ is a.c.c.s.o.f. in $\mathcal{A}$ - if:

1. $\forall m,n \in \mathbb{N}:$ if $ \mathcal{A} \cap X_{m,n} \neq \emptyset \Rightarrow (\mathcal{A} \cap X_{m,n}, +, \cdot)$ is a ring, where $\cdot$ is the usual multiplication, i.e. $(f \cdot g)(x) := f(x) \cdot g(x)$;
2. $\forall g,h \in \mathcal{A}$, with $g:B \rightarrow \mathbb{C}^n$ and $h:C \rightarrow \mathbb{C}, B,C \subseteq \mathbb{C}^m$, and denoted with $f:A \rightarrow \mathbb{C}^n, A:=\{x \in \mathbb{C}^m: g(x) \neq 0 \}, f(x):=\frac{g(x)}{h(x)} \Rightarrow f \in \mathcal{A}$;
3. $\forall n \in \mathbb{N}^+$, $id_{\mathbb{C}^m} \in \mathcal{A}$
4. $\forall i,k \in \mathbb{N}^+$, with $i \leq k-1$, $\tilde{\pi}_{i,k} \in \mathcal{A}$;
5. $\forall m, n \in \mathbb{N}^+$, $\mathcal{i}_{m,n} \in \mathcal{A}$;
6. $\forall g:A \rightarrow \mathbb{C}^n, \forall h:B \rightarrow \mathbb{C}^p, A \subseteq \mathbb{C}^p, B \subseteq \mathbb{C}^m \mbox{ s.t. }Im(h) \subseteq A: \mbox{ if } f,g \in \mathcal{A} \Rightarrow f = g \circ h \in \mathcal{A}$.

It is noted that for any $\mathcal{H} \subseteq X,$ there always exists a set $\mathcal{A},$ with $\mathcal{H}$ being a.c.c.s.o.f. in $\mathcal{A}$, as one can trivially take $\mathcal{A}=X$, which satisfies 1 to 6 above. Therefore, we can define the Algebraic and Compositional Closure of $\mathcal{H}$, $\mathcal{ACC}(\mathcal{H})$, as the intersection of all $\mathcal{A} \subseteq \mathcal{X}$ s.t. $\mathcal{H}$ is a.c.c.s.o.f. in $\mathcal{A}$.

Now, let's consider a subset of polynomials $\emptyset \neq \mathcal{P} \subseteq \mathbb{C}^{\leq n}[x]$, for some $n \in \mathbb{N}$, and some sub-manifolds $A_1, \dots,A_l \subseteq \mathbb{C}^{n+1}$. Let's also denote with $A = \bigcup_{i \in \{1,\dots,l\}}A_i$. we say that $(A_1, \dots,A_l)$ is a Root Generating Function Domain Set for $\mathcal{P}$, in short $\mathcal{H}$ is r.g.f.d.s. for $\mathcal{P}$, if :

$\exists h_1,\dots,h_l \in X^1$, such that dom$(h_i)=A_i, \forall i \in \{1,\dots,l\}$, i.e. $h_i:A_i \rightarrow \mathbb{C}$, $\forall i \in \{1,\dots,l\}$, such that, denoted with $\mathcal{H}=\{h_1,\dots,h_l\} \subseteq X^1 \subseteq X$, we have that:

$\exists f_1,...,f_h \in \mathcal{ACC}(\mathcal{H})$, with $f_i \in X^1 \forall i \in \{1,\dots,h\}$, s.t.:

1. $\forall p \in \mathcal{P}: p(\alpha) = 0 \Rightarrow \alpha \in \{f_1(\pi(p)),\dots,f_h(\pi(p))\}$.
2. $\forall j \in \{1,\dots,h\}, \exists p \in \mathcal{P}$: if $\alpha = f_j(\pi(p)) \Rightarrow p(\alpha)=0$

It is observed, as shown in (*) below, that for any $\emptyset \neq \mathcal{P} \subseteq \mathbb{C}^{\leq n}[x], n \in \mathbb{N}$, there always exists $(A_1,\dots,A_l)$ that is r.g.f.d.s. for $\mathcal{P}$.

We can then define the Minimum Root Function Generator Domain Dimension as the minumum of (the non-empty set) of the maximum across the dimensions of $A_1,\dots,A_l$ s.t. $(A_1,\dots,A_l)$ is r.g.f.d.s. for $\mathcal{P}$:

$MRFGDD(\mathcal{P}) = min\left\{max\left(\mbox{dim}(A_1),\dots,\mbox{dim}(A_l)\right):(A_1,\dots,A_l) \mbox{ is }r.g.f.d.s.\mbox{ for } \mathcal{P} \right\}$.

At this point, two questions:

1. Given $n \in \mathbb{N}^+$, what is $MRFGDD(\mathcal{P})$, when $\mathcal{P}=C^n[x]$?
2. Given $n \in \mathbb{N}^+$, what is $MRFGDD(\mathcal{P})$, when $\mathcal{P}=C^{\leq n}[x]$?

For example, when $n=2, \mathcal{P}=C^2[x]$, one can see that $MRFGDD(\mathcal{P}) \leq 1$ (I guess it is exactly 1, even though it is not immediate to me to show it given the possibility of taking, recursively, compositions of functions).

Indeed, if we take $f:\mathbb{C} \rightarrow \mathbb{C}, f(a_0) = \sqrt{a_0}$ (once chosen a $\sqrt{\cdot}$ function on the all $\mathbb{C}$, also not continuous), then $(A_1)=(\mathbb{C})$ is r.g.f.d.s. for $\mathcal{P}$, with corresponding $h=f=h_1=f_1:\mathbb{C} \rightarrow \mathbb{C}$ and $\mathcal{H}=\{h\}$, as any root of a second-degree polynomial $p$, with $p(x) = a_0+a_1 \cdot x + a_2 \cdot x^2$, can be written as $f_1(a_0,a_1,a_2) = \frac{-a_1+f(a_1^2-4a_2a_0)}{2a_2}$, or $f_2(a_0,a_1,a_2)=\frac{-a_1-f(a_1^2-4a_2a_0)}{2a_2}$, and $f_1=\frac{-\tilde{\pi}_{1,2}+f\circ(\tilde{\pi}_{1,2}^2-4\tilde{\pi}_{2,2}\cdot\tilde{\pi}_{0,2})}{2\tilde{\pi}_{2,2}}, f_2=\frac{-\tilde{\pi}_{1,2}-f\circ(\tilde{\pi}_{1,2}^2-4\tilde{\pi}_{2,2}\cdot\tilde{\pi}_{0,2})}{2\tilde{\pi}_{2,2}}$, so that $f_1, f_2 \in ACC(\mathcal{H})$.

It would be interesting to know what happens for higher values of $n$.

(*)

Here, I show that for any $\emptyset \neq \mathcal{P} \subseteq \mathbb{C}^{\leq n}[x], n \in \mathbb{N}$, there always exists $(A_1,\dots,A_l)$ that is r.g.f.d.s. for $\mathcal{P}$.

Indeed, $\mathcal{P} = \bigcup_{k=1}^{n} \mathcal{P}_k$, where $\mathcal{P}_k := \{f \in \mathcal{P}: \mbox{deg}(p) = k\}$. Now, $\forall p \in \mathcal{P}_k, \exists h_{1,p},...,h_{k,p} \in \mathbb{C}$, s.t. $p(\alpha) = 0 \Leftrightarrow \alpha \in \{h_{1,p},...,h_{k,p}\}$. Then, by the axiom of choice, $\forall k \in \mathbb{N}^+, k \leq n,$ s.t. $\mathcal{P}_k \neq \emptyset: \exists h_{1,k},...,h_{k,k}, h_{i,k}:\mathcal{P}_k \rightarrow \mathbb{C}$, s.t. $p(\alpha) = 0 \Leftrightarrow \alpha \in \{h_{1,k}(p),...,h_{1,k}(p)\}$. If we define $f_{i,k} := h_{i,k} \circ \pi^{-1}_{|\mathbb{C}^k}, f_{i,k}:\mathbb{C}^k \rightarrow \mathbb{C}$, then it suffices to take $\mathcal{H} = vec\left(\{f_{i,k}: i,k \in \mathbb{N}^+, i \leq k \leq n, \mathcal{P}_k \neq \emptyset \}\right) = (h_1,\dots,h_l) = (f_1,\dots,f_h)$ and, correspondingly, $A_j = \mathbb{C}^k$, if $h_j=f_j=f_{i,k}$. We have that:

1. $\forall p \in \mathcal{P}: p \in \mathcal{P}_k, \mbox{ for some } k \in \mathbb{N}^+, k \leq n$, therefore if $p(\alpha) = 0$, then $\alpha \in \{h_{1,p},...,h_{k,p}\} = \{h_{1,k}(p),...,h_{1,k}(p)\} = \{f_{1,k}(\pi(p),...,f_{k,k}(\pi(p))\}$ and each $f_{i,k}=f_{j_{i,k}}$ for some $j_{i,k} \in \{1,...,h\}$.
2. $\forall j \in \{1,...,h\}: f_j = f_{i_j, k_j},$ for some $i_j, k_j \in \mathbb{N}^+, i_j \leq k_j \leq n$. Selected any arbitrary $p \in \mathcal{P}_{k_j}$, which is possible because $\mathcal{P}_{k_j}$ is non-empty by construction, if $\alpha = f_j(\pi(p)) = f_{i_j, k_j}(\pi(p)) = h_{i_j, k_j}(p) \in \{h_{1,p},...,h_{k_j,p}\} \Rightarrow p(\alpha)=0$

Therefore $(A_1,\dots,A_l)$ satisfies points 1,2 and therefore is r.g.f.d.s. for $\mathcal{P}$.