Recently, while studying certain notions of "averaging" a set of input matrices, I obtained a nonlinear polynomial in matrix variables. A simple example is

\begin{equation*}
\mathcal{G}(X) := X^n - \sum_{i=0}^{n-1} (C_iX^i + X^iC_i),
\end{equation*}

where the $C_i$ are symmetric positive definite matrices. 

If all the terms above were scalars, then Descartes' Rule of Signs tells us that $\mathcal G$ has exactly one positive root. This led me to wonder if a similar "rule" is also known for the above case.

> Does there exist a unique symmetric positive definite solution to $\mathcal{G}(X)=0$? 

*Note:* Obviously, if the $C_i$ commute with each other, we can simultaneously diagonalize everything and apply the rule of signs to conclude uniqueness. I'm hoping the same conclusion also holds for the general case.