The Complexity of the Minimal Polynomial

Thanh Minh Hoang and Thomas Thierauf

We investigate the computational complexity of the minimal polynomial of an integer matrix.

We show that the computation of the minimal polynomial is in AC0(GapL), the AC0-closure of the logspace counting class GapL, which is contained in NC2. Our main result is that the problem is hard for GapL (under AC0 many-one reductions). The result extends to the verification of all invariant factors of an integer matrix.

Furthermore, we consider the complexity to check whether an integer matrix is diagonalizable. We show that this problem lies in AC0(GapL) and is hard for AC0(C=L) (under AC0 many-one reductions).