I've been waiting until our paper was published to answer this one, and it finally was. We give a formal power series solution to the general polynomial.
Let's start here with the solution to a polynomial in what we call geometric form, meaning a constant of $1$ and a linear coefficient of $-1$.
Theorem The polynomial or power series equation
$$0= 1 - \alpha + t_2 \alpha ^2 + t_3 \alpha^3 +t_4 \alpha^4 + \ldots $$
has a formal power series solution:
$$ \alpha %={\mathbf S}= {\mathbf S}[t_2,t_3,t_4,\ldots]
= \sum_{m_2,m_3,m_4,\ldots \ge 0} \! C[m_2,m_3,m_4,\ldots] \ t_2^{m_2} t_3^{m_3} t_4^{m_4} \cdots \equiv \sum_{{\mathbf m} \ge {\mathbf 0}} C_{\mathbf m} {\mathbf t}^{\mathbf m} $$
where the hyper-Catalan number $C_{\mathbf m}=C[m_2, m_3, m_4, \ldots]$ counts the number of roofed polygons subdivided with non-crossing diagonals into $m_2$ triangles, $m_3$ quadrilaterals, $m_4$ pentagons, etc.
The closed-form for $C_{\mathbf m}$ has been known since Erdelyi and Etherington found it in 1941:
$$C_{\mathbf m} = \dfrac{( 2m_2 + 3m_3 + 4m_4 + \ldots )!}{(1 + m_2 + 2m_3 + 3m_4 + \ldots)! \, m_2! \, m_3! \, m_4! \cdots}
$$
That can be written as:
$$C_{\mathbf m}= \dfrac{( E_{\mathbf m}-1)!}{ (V_{\mathbf m} -1) ! \, {\mathbf m}! \ }$$
where $V_{\mathbf m}=2+ m_2 + 2m_3 + 3m_4 + \ldots$ and $E_{\mathbf m} = 1 +2m_2 + 3m_3 + 4m_4 + \ldots $ are the number of vertices and edges of a roofed subdivided polygon of type ${\mathbf m}$ and ${\mathbf m}! \equiv m_2! m_3! m_4! \cdots$.
A straightforward substitution yields the general polynomial formula:
Theorem The polynomial or power series equation
$$ \displaystyle 0 = c_0 - c_1 \alpha + c_2 \alpha^2 + c_3 \alpha^3 + c_4 \alpha^4 + \ldots
$$
has a formal series solution:
\begin{align*}
\alpha & =
\!\!\! \!\!\! \sum_{\substack{m_2, m_3, \ldots \ge 0 \ \ }} \!\! \!\!
\dfrac{( 2m_2 + 3m_3 + 4m_4 + \ldots )!}{(1 + m_2 + 2m_3 + 3m_4 +\ldots)! m_2! m_3! \cdots} \ \dfrac{ c_0^{1 + m_2 + 2m_3 +\ldots} c_2^{m_2 } c_3^{m_3}\cdots }
{c_1^{1 + 2m_2 + 3m_3 + 4m_4 + \ldots}}
\\
& = \sum_{{\mathbf m} \ge {\mathbf 0}} \ \dfrac { c_0^{V_{\mathbf m} -1}}{(V_{\mathbf m} -1) !} \ \dfrac{( E_{\mathbf m}-1)!} { c_1^{E_{\mathbf m}}} \ \dfrac {{\mathbf c}^{\mathbf m}}{{\mathbf m}! } .
\end{align*}
where ${\mathbf c}^{\mathbf m} \equiv c_2^{m_2} c_3^{m_3} c_4^{m_4} \cdots$.
Reference: Wildberger, N. J., & Rubine, D. (2025). A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode. The American Mathematical Monthly, May 2025. https://doi.org/10.1080/00029890.2025.2460966
