Computations associated with the resonance arrangement

TL;DR

This paper computes the characteristic polynomial and Betti numbers of the resonance hyperplane arrangement for n=9, providing new explicit formulas and advancing understanding of its algebraic and topological properties.

Contribution

The authors compute the characteristic polynomial and Betti numbers for the resonance arrangement at n=9, introducing new explicit formulas for higher Betti numbers using computational methods.

Findings

Computed the characteristic polynomial for n=9

Derived explicit formula for Betti number b_4

Enhanced understanding of the arrangement's algebraic structure

Abstract

The resonance arrangement $\mathcal{A}_n$ is the arrangement of hyperplanes in $\mathbb{R}^n$ given by all hyperplanes of the form $\sum_{i \in I} x_i = 0$, where $I$ is a nonempty subset of $\{1,\dots,n\}$. We consider the characteristic polynomial $\chi(\mathcal{A}_n; t)$ of the resonance arrangement, whose value $R_n$ at $-1$ is of particular interest, and corresponds to counts of generalized retarded functions in quantum field theory, among other things. No formula is known for either the characteristic polynomial or $R_n$, though $R_n$ has been computed up to $n=8$. By exploiting symmetry and using computational methods, we compute the characteristic polynomial of $\mathcal{A}_9$, and thus obtain $R_9$. The coefficients of the characteristic polynomial are also equal to the so-called Betti numbers of the complexified hyperplane arrangement; that is, the coefficient of $t^{n-i}$ is denoted by the Betti number $b_i(\mathcal{A}_n)$. Explicit formulas are known for the Betti numbers up to $b_3(\mathcal{A}_n)$. Using computational methods, we also obtain an explicit formula for $b_4(\mathcal{A}_n)$, which gives the $t^{n-4}$ coefficient of the characteristic polynomial.