Solving Via Eigenvalues: a Singular Implementation

Systems of polynomial equations play an important role in many scientific applications. But it is often rather complex and time-consuming to find all -real and complex- solutions. This book describes an efficient algorithm, which uses eigenvalues to compute all solutions of a given system of polynomial equations. For this, the theory of Gröbner bases is combined with numerical linear algebra. Also, a comparison to the performance of existing algorithms is given. Furthermore, a new algorithm to compute the primary decomposition of a zero-dimensional ideal and an algorithm to compute the number of real respectively complex roots of a system of polynomial equations using the quadratic form is delineated. All described algorithms are implemented in the computer algebra system SINGULAR.