The Coboundary Quasi-Polynomials of Hyperplane Arrangements over Residually Finite Dedekind Domains

TL;DR

This paper introduces a new polynomial invariant that refines both the coboundary polynomial and the characteristic quasi-polynomial for hyperplane arrangements over residually finite Dedekind domains, unifying previous concepts.

Contribution

It presents a novel common refinement of the coboundary polynomial and characteristic quasi-polynomial specifically for arrangements over residually finite Dedekind domains.

Findings

Defines the common refinement polynomial for such arrangements.

Shows the polynomial generalizes existing invariants.

Provides properties and potential applications of the new polynomial.

Abstract

The characteristic polynomial plays an important role in study of hyperplane arrangements. There are several refinements of the characteristic polynomial. One of them is the coboundary polynomial defined by Crapo. Another refinement is the characteristic quasi-polynomial for an integral arrangement defined by Kamiya, Takemura, and Terao. Recently, the first and third authors introduced the characteristic quasi-polynomial for arrangement defined over a residually finite Dedekind domain. In this article, we introduce the common refinement of the coboundary polynomial and characteristic quasi-polynomial for an arrangement over a residually finite Dedekind domain.