Overview
This work reformulates Zaslavsky's theorem, which enumerates the number of regions at a specific level, $r_\ell(\mathcal{A})$, within a real hyperplane arrangement $\mathcal{A}$. The reformulation is presented in terms of a construction referred to as the centralization of $\mathcal{A}$. A bijective proof for this restated theorem is provided. The research then applies this framework to address questions concerning the concept of 'level' in two distinct areas: generalizing the enumeration to geometric semilattices and connecting it to existing results on level-related phenomena.
Research Context
Zaslavsky's prior work detailed the computation of $r_\ell(\mathcal{A})$, which refines his established methods for enumerating total regions and relatively bounded regions of a hyperplane arrangement. Despite the existence of Zaslavsky's level-counting theorem, recent investigations into the phenomenon of level have not extensively utilized it. The present study seeks to apply this theorem to obtain or generalize many of the findings from these recent investigations.
Approach
The research initiates by restating Zaslavsky's theorem on counting regions of a real hyperplane arrangement based on their level. This restatement is framed using the concept of the centralization of $\mathcal{A}$. A bijective proof is then constructed to validate this reformulation. Following the proof, the framework is applied in two principal directions.
- The first application explores the implication that $r_\ell(\mathcal{A})$ is determined solely by the intersection poset $\mathcal{L}(\mathcal{A})$. This dependency allows for the definition of both $r_\ell$ and centralization within the broader mathematical context of geometric semilattices. Within this generalized setting, a very general expression for the characteristic polynomial of a geometric semilattice is derived, along with several corollaries.
- The second application focuses on integrating the restated theorem with recent research into the phenomenon of level. The study demonstrates how Zaslavsky's level-counting theorem can be used to derive or generalize existing results. Specifically, it shows how exponential generating function identities, as presented in arXiv:2410.10198 and arXiv:2411.02971, and an expression for the characteristic polynomial in terms of $r_\ell$, found in arXiv:2411.03756, can be derived for deformations of the braid arrangement.
Findings
- Zaslavsky's theorem for computing $r_\ell(\mathcal{A})$ can be restated and rigorously proven using the construction of the centralization of $\mathcal{A}$.
- The enumeration indicated by $r_\ell(\mathcal{A})$ depends exclusively on the intersection poset $\mathcal{L}(\mathcal{A})$. This functional dependence allows for the extension of both $r_\ell$ and the concept of centralization to geometric semilattices.
- Within the context of geometric semilattices, a general expression for their characteristic polynomial was derived, accompanied by several corollaries.
- The established level-counting theorem by Zaslavsky is applicable to recent work concerning the phenomenon of level, facilitating the derivation or generalization of their outcomes.
- Specific application showed that exponential generating function identities from arXiv:2410.10198 and arXiv:2411.02971, and an expression for the characteristic polynomial in terms of $r_\ell$ from arXiv:2411.03756, can be derived for deformations of the braid arrangement using this reinterpreted theorem.