UC Berkeley

In geometric, algebraic, and topological combinatorics, properties such as symmetry, unimodality, and real-rootedness of combinatorial generating polynomials are frequently studied. In this thesis, we present three projects exploring various properties of polynomials arising from Ehrhart theory, the study of counting integer points in lattice polytopes. Many of the open questions on real-rootedness and unimodality of polynomials pertain to the enumeration of faces of cell complexes. When proving that a polynomial is real-rooted, we often rely on the theory of interlacing polynomials and their recursive nature. 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.

The Ehrhart polynomial $\text{ehr}_P(t)$ of a lattice polytope $P$ counts the number of integer points in the $n$-th dilate of $P$. The $f^*$-vector of $P$, introduced by Felix Breuer in 2012, is the vector of coefficients of $\text{ehr}_P(n)$ with respect to the binomial coefficient basis $ \left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}$, where $d = \text{dim} P$. Similarly to $h/h^*$-vectors, the $f^*$-vector of $P$ coincides with the $f$-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of $f^*$-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of $f$-vectors of simplicial polytopes; e.g., the first half of the $f^*$-coefficients increases and the last quarter decreases. Even though $f^*$-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart $h^*$-vector, there is a polytope with the same $h^*$-vector whose $f^*$-vector is unimodal.

Posets can be viewed as subsets of the type-A root system that satisfy certain properties. Geometric objects arising from posets, such as order cones, order polytopes, and chain polytopes, have been widely studied, though many open questions concerning them remain open, such as the unimodality of their $h^*$-vectors. In 1993, Vic Reiner introduced signed posets, which are subsets of the type-B root system that satisfy the same properties. We introduce the analogue of order and chain polytopes in this setting, focusing on the Ehrhart theory of these objects. We are able to determine when these signed order polytopes have symmetric $h^*$-vectors, and end with a discussion of open questions regarding signed chain polytopes.