Computing singular solutions to polynomial systems

Authors:
Alexander P Morgan;Andrew J Sommese;Charles W Wampler
Affiliations:
Mathematics Department, General Motors Research Laboratories, Warren, Michigan 48090 USA;Mathematics Department, University of Notre Dame, Notre Dame, Indiana 46556 USA;Mathematics Department, General Motors Research Laboratories, Warren, Michigan 48090 USA
Venue:
Advances in Applied Mathematics
Year:
1992

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Abstract

A method to generate accurate approximations to the singular solutions of a system of (complex) polynomial equations is presented. This method is established in a context of polynomial continuation; thus, all solutions are generated, with the singular solutions being approximated more accurately than by standard implementations. The theorem on which the method is based is proven using results from several complex variables and algebraic geometry. No special conditions on the derivatives of the system, such as restrictions on the rank of the Jacobian matrix at solutions, are required. A specific implementation is given and the results of numerical experiments in solving four test problems are presented.