Decoupling Highly Structured Polynomial Systems

An efficient technique for solving polynomial systems with a particular structure is presented. This structure is very specific but arises naturally when computing the critical points of a symmetric polynomial energy function. This novel numerical solution method is based on homotopy continuation and is a particular application of results due to Canny and Rojas. An illustrative example from magnetism is presented.

[1]  J. Yorke,et al.  The cheater's homotopy: an efficient procedure for solving systems of polynomial equations , 1989 .

[2]  A. Morgan,et al.  Coefficient-parameter polynomial continuation , 1989 .

[3]  Chris Peterson,et al.  Perturbed Regeneration for Finding All Isolated Solutions of Polynomial Systems , 2013 .

[4]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[5]  Andrew J. Sommese,et al.  The numerical solution of systems of polynomials - arising in engineering and science , 2005 .

[6]  Tsung-Lin Lee,et al.  HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method , 2008, Computing.

[7]  Jonathan D. Hauenstein,et al.  Numerically Solving Polynomial Systems with Bertini , 2013, Software, environments, tools.

[8]  Jan Verschelde,et al.  Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation , 1999, TOMS.

[9]  Wenrui Hao,et al.  Cell Cycle Control and Bifurcation for a Free Boundary Problem Modeling Tissue Growth , 2013, Journal of Scientific Computing.

[10]  T. Y. Li.  Numerical solution of multivariate polynomial systems by homotopy continuation methods , 2008 .

[11]  A. J. Newell,et al.  Transition to superparamagnetism in chains of magnetosome crystals , 2009 .

[12]  J. Maurice Rojas,et al.  An optimal condition for determining the exact number of roots of a polynomial system , 1991, ISSAC '91.

[13]  Jonathan D. Hauenstein,et al.  Regeneration homotopies for solving systems of polynomials , 2010, Math. Comput..