论文标题
Macaulay2中的懈怠理想
Slack Ideals in Macaulay2
论文作者
论文摘要
最近，Gouveia，Thomas和作者引入了Slack实现空间，这是一个新模型，用于实现多层的实现空间。它通过其松弛矩阵代表每个多层，该基质是通过评估每个顶点处的每个刻面不等式而获得的。与经典模型不同，Slack模型自然会模拟投影转换。它固有地是代数，是各种饱和决定性理想的积极部分，并提供了一种新的计算工具来研究多元化的经典可实现性问题。我们介绍了Macaulay2的软件包，该软件包提供了创建和操纵凸多属和矩阵的松弛矩阵和松弛理想的方法。松懈的理想通常很难计算。为了提高Slack模型的功能，我们开发了两种策略来简化计算：我们将Slack矩阵的尽可能多的条目扩展到一个；然后，我们获得了一个减少的松弛模型，该模型将松弛品种与更紧凑的Grassmannian实现空间模型相结合。这使我们能够研究以前无法计算的理想。作为应用，我们表明，众所周知的Perles Polytope不承认合理的实现，并证明了大型准模拟球体的不实现性。
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 quasi-simplicial sphere.
