For real 3x3 matrices X, the set of solutions to X^n + a_{n-1}X^{n-1} + ... + a_0 I = O is a union of similarity orbits whose covering dimension is always 0, 4, or 6, or else the set is empty.
Gelfand, On the number of solutions of a quadratic equation, In: Globus: General Mathematical Seminar, 1, Independent University of Moscow, Moscow, 2004, 124-133 (in Russian)
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Simple polynomial equations over (mxm)-matrices
For real 3x3 matrices X, the set of solutions to X^n + a_{n-1}X^{n-1} + ... + a_0 I = O is a union of similarity orbits whose covering dimension is always 0, 4, or 6, or else the set is empty.