Are there some systematic approaches to find and/or describe solution spaces for polynomial equations involving matrices?
I am especially interested in equations of the type $${\bf X}^{N} = {\bf A}$$
For example when $\bf A$ represents a motion of some kind. Possibly affine transformations and geometrical projections.
I am able to numerically solve such equations by using smoothed fixed point methods. But this only gives me singular solutions and only depending on an initial guess. I suspect that there could exist large sets of solutions which could live in some subset of matrices ${\bf X} \in \mathcal X$. Especially interesting are solution spaces in which $\bf X$ are close to $\bf I$ in some sense. In my mind this should ensure a smooth sequence ${\bf X},{\bf X}^2, \cdots, {\bf X}^N$ which is of interest to my applications which mostly reside in the realms of computer graphics, image processing and video coding.