Toric eigenvalue methods for solving sparse polynomial systems

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.

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techReport Toric eigenvalue methods for solving sparse polynomial systems S.J.L. Telen (Simon Jo L) and M.R. Bender (Matias)