Polyhedral homotopies in Cox coordinates

Homotopy continuation is an important technique for solving systems of polynomial equations numerically. A basic approach is to track the Bézout number many paths in an affine space. More advanced strategies include polyhedral homotopies and (multi) projective homotopies. The former exploits the polyhedral structure of the equations to reduce the number of paths, while the latter avoids diverging paths by tracking in a compact space. We combine the advantages of these two strategies by tracking paths in a compact toric variety, naturally associated to our polynomial system. A quotient construction by Cox allows us to use global coordinates on this toric variety. I will explain the main ideas and illustrate the advantages of our approach by means of examples. This is joint work with Timothy Duff, Elise Walker and Thomas Yahl.