Authors: 

Joseph Cummings (University of Edinburgh)

Jonathan D. Hauenstein (University of Notre Dame) [speaker]

Title: 

Mixed volume and ramification points of polyhedral homotopies

Abstract:

Numerical algebraic geometry utilizes homotopy continuation to solve polynomial systems via numerically tracking solution paths.  The homotopies of interest here
are polyhedral homotopies which arise between two polynomial systems with the same Newton polytopes.  A singularity along a solution path of a homotopy is a ramification point.  Building on previous work of ramification points of homotopies, we develop new results showing that the number of ramification points for polyhedral homotopies can be described in terms of mixed volumes.  This interpretation utilizes a nef divisor which is shown to be determined by solving a linear program.  Software combining the computation of mixed volumes and solving a linear program used for performing such computations has been developed using Macaulay2 and Python, and has been used with several examples to demonstrate the new results.