Are there interesting relations between the coefficients of the characteristic polynomial of a companion matrix and the coefficients of (some of) the characteristic polynomials of its (matrix) powers?
I am not searching for the trivial ones given in relation to the roots of these polynomials or a decomposition in irreducible factors as in general I cannot ensure the existence of these objects for all of my rings. I only have access to the coefficients.
If $P(x) = \sum_{i=0}^n c_i x^n$ is the characteristic polynomial of $A$, then $(-1)^n P(\sqrt{x}) P(-\sqrt{x})$ is the characteristic polynomial of $A^2$. Thus the coefficient of $x^k$ in the characteristic polynomial of $A^2$ is $$ (-1)^n \sum_{j = \max(0,2k-n)}^{\min(n,2k)} (-1)^j c_j c_{2k-j} $$ Something similar should be true for the $p$'th power, using $\prod_{\omega} P(\omega x^{1/p})$ where the product is over all $p$'th roots of unity.