Slack Ideals in Macaulay2
Slack Ideals in Macaulay2
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DOI：
10.1007/978-3-030-52200-1_22
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发表时间：
2020-06-06
期刊：
影响因子：
--
通讯作者：
Wiebe A
中科院分区：
文献类型：
--
作者：
Macchia A;Wiebe A
最近Gouveia、Thomas等人提出了一种新的多面体实现空间模型——松弛实现空间。它用松弛矩阵表示每个多面体，松弛矩阵是通过计算每个顶点上的每个面不等式得到的。与经典模型不同，松弛模型自然地模化了射影变换。它本质上是代数的，是各种饱和行列式理想的正部分，为研究经典多面体的可实现性问题提供了一种新的计算工具。我们介绍了用于Macaulay2的软件包SlackIdeals，它提供了创建和操作松弛矩阵以及凸多边形和拟阵的松弛理想的方法。松弛的理想往往难以计算。为了提高松弛模型的功率，我们开发了两种策略来简化计算：我们将尽可能多的松弛矩阵的条目缩放为1；然后将松弛变化与更紧凑的格拉斯曼实现空间模型相结合，得到一个简化的松弛模型。这使我们能够研究以前计算无法达到的松弛理想。作为应用，我们证明了众所周知的Perles多面体不允许理性实现，并证明了大简单球的非可实现性。
Recently Gouveia, Thomas and the authors introduced the slack realization space, a new model for the realization space of a polytope. It represents each polytope by its slack matrix, the matrix obtained by evaluating each facet inequality at each vertex. Unlike the classical model, the slack model naturally mods out projective transformations. It is inherently algebraic, arising as the positive part of a variety of a saturated determinantal ideal, and provides a new computational tool to study classical realizability problems for polytopes. We introduce the package SlackIdeals for Macaulay2, that provides methods for creating and manipulating slack matrices and slack ideals of convex polytopes and matroids. Slack ideals are often difficult to compute. To improve the power of the slack model, we develop two strategies to simplify computations: we scale as many entries of the slack matrix as possible to one; we then obtain a reduced slack model combining the slack variety with the more compact Grassmannian realization space model. This allows us to study slack ideals that were previously out of computational reach. As applications, we show that the well-known Perles polytope does not admit rational realizations and prove the non-realizability of a large simplicial sphere.
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影响因子：
0.8
作者：
Gouveia, Joao;Macchia, Antonio;Wiebe, Amy
通讯作者：
Wiebe, Amy
影响因子：
1.1
作者：
YANNAKAKIS, M
通讯作者：
YANNAKAKIS, M
影响因子：
1.1
作者：
Gouveia, Joao;Pashkovich, Kanstanstin;Thomas, Rekha R.
通讯作者：
Thomas, Rekha R.
影响因子：
0.8
作者：
Gouveia, Joao;Macchia, Antonio;Wiebe, Amy
通讯作者：
Wiebe, Amy