Toric eigenvalue methods for solving sparse polynomial systems

We consider the problem of computing homogeneous coordinates points in a zero-dimensional subscheme compact, complex toric variety X X . Our starting point is ideal I"> I encoding="application/x-tex">I Cox ring , which practice might arise from homogenizing sparse polynomial system. prove new eigenvalue theorem compact setting, leads to novel, robust numerical approach for solving this problem. method works particular systems having isolated solutions with arbitrary multiplicities. It depends on multigraded regularity properties study these and provide bounds size matrices our when complete intersection.