Toric eigenvalue methods for solving sparse polynomial systems
Mathematical Computation , Volume 91 p. 2397- 2429
We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety . Our starting point is a homogeneous ideal in the Cox ring of , which in practice might arise from homogenizing a sparse polynomial system. We prove a new eigenvalue theorem in the toric compact setting, which leads to a novel, robust numerical approach for solving this problem. Our method works in particular for systems having isolated solutions with arbitrary multiplicities. It depends on the multigraded regularity properties of . We study these properties and provide bounds on the size of the matrices in our approach when is a complete intersection.