Computing Primary and Maximal Components

Using the connection between generalized eigenspaces of the multiplication family and primary components of a zero-dimensional polynomial ideal \(I\), we examine various methods for computing the primary and maximal components of \(I\) in the fifth chapter. In addition to the generic approach, i.e., the approach using a generic linear form to get a splitting endomorphism, we use random linear forms, the power of the Frobenius endomorphism over finite base fields, repeated factorization over field extensions, the calculation of the maximal components from the radical ideal, and the computation of the primary components from the maximal components. The chapter finishes with some recent insights into the calculation of the separable subalgebra.