On polynomial values of the discriminants of characteristic polynomials (Q1352642)

For a square matrix \(A\), denote by \({\mathcal D} (A)\) the discriminant of its monic characteristic polynomial. Under some necessary conditions imposed on \(A\) and \(f\), \textit{J. G. Grytczuk} [Discuss. Math. 12, 45-51 (1992; Zbl 0787.11004)] showed that if \(A\) is a \(2\times 2\)-matrix with entries in \({\mathbb{Z}}\) and if \(f(X)\) is a monic polynomial in \({\mathbb{Z}}[X]\), then the equation (*) \({\mathcal D} (A^n) =f(x)\) has only finitely many equations in \(x\in{\mathbb{Z}}\), \(n\in{\mathbb{N}}\) which can all be effectively determined in terms of \(A\) and \(f\). The authors consider equation (*) in the more general situation that \(A\) is a matrix of dimension \(l\geq 2\) with entries in \(R\) and that \(f(X)\in R[X]\), where \(R\) is a finitely generated subring of \({\mathbb{C}}\). Moreover, the authors allow \(l\), \(A\), \(x\) and \(n\) to be unknowns. Their main result is that if \(R\) is effectively represented, and \(f(X)\in R[X]\) is a fixed polynomial with at least three zeros of odd multiplicity, then (*) has only finitely many solutions \((l,A,x,n)\) such that \(l\geq 2\), \(A\) is a non-trivial \(l\times l\)-matrix with eigenvalues in \(R\), \(x\in R\), \(n\in{\mathbb{N}}\), and moreover, that these solutions can be determined effectively in terms of \(f\) and \(R\). The notion of non-triviality of a matrix is too complicated to explain here, but for instance a matrix is non-trivial if it is non-singular and if no quotient of two of its eigenvalues is a root of unity. The authors use the fact that \({\mathcal D} (A^n)\) is a square in \(R\) if the eigenvalues of \(A\) belong to \(R\), and apply an effective finiteness result of Brindza on the solutions of the hyperelliptic equation \(y^2=f(x)\) in \(x,y\in R\).