2020/12/08 by Timothy Duff, Duff, Timothy, Simon Telen +5
Mathematics · Computer Science · Physics and Astronomy · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2012.04255
We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety XΣ. The algorithm lends its name from a construction, described by Cox, of XΣ as a GIT quotient XΣ= (ℂk ∖ Z) // G of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space ℂk of XΣ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of XΣ. It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the G-orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of XΣ. In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound.