论文标题
用于求解稀疏多项式系统的复曲面特征值方法
Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
论文作者
论文摘要
我们考虑了计算零维的零级副本的均匀坐标的问题。我们的起点是$ x $的Cox环中的一个均匀的理想$ i $，实际上，这可能是由于稀疏多项式系统均质而产生的。我们证明了在感谢您的紧凑型环境中的一种新的特征值定理，这导致了一种新颖的，可靠的数值方法来解决此问题。我们的方法特别适用于具有任意多重性的隔离解决方案的系统。这取决于$ i $的多面规律性属性。我们研究这些属性，并在$ i $是完整的交叉点时就矩阵出现的矩阵大小提供界限。
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 appearing in our approach when $I$ is a complete intersection.
