Moderator Note: This question is from a contest which ended 1 Dec 2012.
Consider a polynomial $f$ with complex coefficients. Call such $f$ broken if we can find a square matrix $M$ such that $M \neq f(N)$ for any square matrix $N$.
My professor told me it was possible to explicitly characterize all broken polynomials of any degree. What would such a characterization look like?
Thank you!