论文标题
可壳式复合物的细分
Subdivisions of Shellable Complexes
论文作者
论文摘要
在几何，代数和拓扑组合学中，经常研究组合产生多项式的非兴趣性。当多项式（真实）稳定时，单形态会遵循，这是通过隔离多项式理论推导的特性。关于多项式的稳定性和非兴趣的许多开放问题与细胞复合物面部枚举的枚举有关。 在本文中，我们将交织多项式的理论与细胞复合物的可壳性相关联。我们首先得出了足够的条件，可以使$ h $ - 多种物质的稳定性的稳定性。为了应用它，我们概括了互相结构域的凸面嵌入多型的凸层以抽象多面体，并使用此概括来定义多面体复合物的稳定壳的家族。我们表征了立方体和简单络合物的稳定炮击，并应用该理论来回答Brenti和Welker的问题，这些问题是众所周知的立方多面体的Barycentric Subdivisions。我们还为穆罕默迪（Mohammadi）和韦尔克（Welker）问题提供了积极的解决方案，这些问题在细胞复合物的边缘细分方面。最后，我们将稳定线炮击的家族与超平面布置的组合联系起来。我们提出了相关的问题，答案将解决一些长期存在的问题，同时加强了交错的多项式理论与超平面布置的组合之间的联系。
In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper, we relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the $h$-polynomial of a subdivision of a shellable complex. To apply it, we generalize the notion of reciprocal domains for convex embeddings of polytopes to abstract polytopes and use this generalization to define the family of stable shellings of a polytopal complex. We characterize the stable shellings of cubical and simplicial complexes, and apply this theory to answer a question of Brenti and Welker on barycentric subdivisions for the well-known cubical polytopes. We also give a positive solution to a problem of Mohammadi and Welker on edgewise subdivisions of cell complexes. We end by relating the family of stable line shellings to the combinatorics of hyperplane arrangements. We pose related questions, answers to which would resolve some long-standing problems while strengthening ties between the theory of interlacing polynomials and the combinatorics of hyperplane arrangements.
