We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, 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 $I$. We study these properties and provide bounds on the size of the matrices involved in our approach in the case where $I$ is a complete intersection.

翻译：我们考虑的是在一个集约的、复杂的氧化物型的零维子系统中计算各点的同质坐标的问题。我们的出发点是Cox环中一个单一的理想美元,即X美元,这实际上可能来自一个稀有的多元系统同质化。我们证明,在Toric紧凑环境中,一种新的精华理论价值,这导致了一种解决该问题的新颖的、稳健的数字方法。我们的方法特别适用于具有独立的多功能解决方案的系统。它取决于多级常规性的$。我们研究这些特性,并在美元是一个完整的交叉点的情况下,提供我们方法所涉及的矩阵大小的界限。