Toric Newton Method for Polynomial Homotopies

This paper deenes a generalization of Newton's method to deal with solution paths de-ned by polynomial homotopies that lead to extremal values. Embedding the solutions in a toric variety leads to explicit scaling relations between coeecients and solutions. Toric Newton is a symbolic-numeric algorithm where the symbolic pre-processing exploits the polyhedral structures. The numerical stage uses the additional variables introduced by the homogenization to scale the components of the solution vectors to the complex unit circle. Toric Newton generates appropriate aane charts and enables to approximate the magnitude of large solutions of polynomial systems.