Using symmetries in the eigenvalue method for polynomial systems
Let \(I\subset \mathbb{C}[x_1, \dots, x_n]\) be a zero--dimensional ideal. For \(f\in\mathbb{C}[x_1, \dots, x_n]\) denote by \(m_f\) the matrix defined by the multiplication with \(f\) in the algebra \(\mathbb{C}[x_1, \dots, x_n]/I\) and \(h_f\) its minimal polynomial. It is known that the roots of \(h_f=0\) coincide with the eigenvalues of \(m_f\) resp. the values of \(f\) on \(V(I)\). Using this fact one can compute the eigenvalues numerically and find the points in \(V(I)\), i.e. the solutions of the polynomial equations defining \(I\). This so-called eigenvalue method is studied and adapted to the case that the system of polynomial equations has symmetries. It is proved that the corresponding matrices occuring in this situation have a special block diagonal structure. An algorithm is given to exploit this result in an efficient way.
scientific article; zbMATH DE number 5611764
scientific article