Rational solutions of the matrix equation \(p(X) = A\) (Q6628864)

Given a polynomial \(p(\lambda)\) with rational coefficients, and given an \(n \times n\) rational matrix \(A\), this work seeks to identify conditions, based on the characteristic polynomial \(f(\lambda)\) of \(A\) and the polynomial \(p(\lambda)\) under which a rational solution a rational solution \(X\) exists for the matrix equation \(p(X)=A\). One such a condition is mentioned in a paper by \textit{R. Reams} [Linear Algebra Appl. 258, 187--194 (1997; Zbl 0884.15008)]. This work extends Theorem 1 from R. Reams' paper, generalizing the analysis beyond the case \(p(\lambda) = \lambda^m\) with odd \(m\) is considered. The paper concludes by proposing a broader problem: given a rational polynomial \(R(x; y)\) in two variables, can we find a pair of rational matrices \((X; Y )\) such that \(R(X; Y ) = 0\)?