论文标题
Cox坐标中的多面体同位素
Polyhedral Homotopies in Cox Coordinates
论文作者
论文摘要
我们介绍了COX同型算法，用于在紧凑的紫红色品种上求解稀疏的多项式方程系统$x_σ$。该算法从Cox描述的结构中借出了$x_σ$的名称，作为git商$x_σ=（\ m athbb {c}^k \ setminus z）// quasi-affine品种的g $通过还原组的作用。我们的算法在总坐标空间中跟踪路径$ \ mathbb {c}^k $的$x_σ$，并且可以看作是标准多面体同型的同质版本，该版本在$x_σ$的密集圆环上起作用。此外，它概括了（多）投影空间中常用的路径跟踪算法，因为它跟踪了与每个解决方案相对应的$ G $ -Orbit中包含的一组均匀坐标。 Cox同拷贝结合了多面体同型和（多）同质同质的优点，仅跟踪混合量的许多解决方案，并提供了一种优雅的方式来处理$x_σ$的特殊部门或附近的解决方案。此外，该策略可能有助于了解某些与BKK结合的系统家族的根数不足。
We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety $X_Σ$. The algorithm lends its name from a construction, described by Cox, of $X_Σ$ as a GIT quotient $X_Σ= (\mathbb{C}^k \setminus Z) // G$ of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space $\mathbb{C}^k$ of $X_Σ$ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of $X_Σ$. It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the $G$-orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of $X_Σ$. In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound.
