Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems
The paper deals with the eigenproblem approach for solving systems of polynomial equations. In particular, the case of isolated zeros only is considered. From a Gröbner basis of the zero-dimensional ideal generated by the polynomial equations an \(n\times n\)-matrix \(A_f\), depending on some suitable polynomial \(f\), can be computed. Here \(n\) denotes the number of zeros counted with multiplicity. The case of simple zeros only, which is shortly reviewed, leads to the classical eigenvalue/eigenvector problem for \(A_f\). The intention of the present paper is to extend the theory to the case of multiple zeros. It results in a generalization of the Jordan normal form decomposition of \(A_f\). The case of nonderogatory matrices \(A_f\) is specially addressed. An algorithmic analysis of the eigenproblem in the general case is indicated.
scientific article; zbMATH DE number 770162