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.

在几何、代数和拓扑组合学中，经常研究组合生成多项式的单峰性。当多项式是（实数）稳定时，就会出现单峰性，这是一种通常通过隔行多项式理论推导出来的性质。许多关于多项式的稳定性和单峰性的悬而未决的问题都与细胞复合体的面的枚举有关。在本文中，我们将交错多项式的理论与细胞复合体的可壳性联系起来。我们首先推导出h稳定的充分条件-可壳复合体细分的多项式。为了应用它，我们将多面体凸嵌入的互易域的概念概括为抽象多面体，并使用这种概括来定义多面体复合体的稳定脱壳族。我们描述了立方和单纯复形的稳定壳，并应用这个理论来回答布伦蒂和韦尔克关于众所周知的立方多胞体的重心细分的问题。我们还对 Mohammadi 和 Welker 关于细胞复合体的边缘细分问题给出了积极的解决方案。我们最后将稳定线壳系列与超平面排列的组合学联系起来。我们提出相关问题，