Characteristic Polynomials of Hyperplane Arrangements via Finite Fields and Tutte Specialization — E8 Intelligence Research

Abstract

FINDING: The characteristic polynomial of a hyperplane arrangement is computed via finite field methods, and its coefficients encode the arrangement's combinatorial structure; the Tutte polynomial of a symmetric arrangement specializes to this characteristic polynomial, linking root systems to chromatic and flow polynomials. MATH: For an arrangement \(\mathcal{A}\) in \(\mathbb{F}_q^d\), the characteristic polynomial \(\chi_{\mathcal{A}}(q) = \sum_{X \in L(\mathcal{A})} \mu(0,X) q^{\dim X}\) (Möbius inversion on the intersection lattice). Finite field method: \(\chi_{\mathcal{A}}(q) = \#(\mathbb{F}_q^d \setminus \bigcup_{H \in \mathcal{A}} H)\) for large primes \(q\). For a Weyl group \(W\) acting on root system \(\Phi\), the arrangement of reflecting hyperplanes has \(\chi_{\mathcal{A}}(t) = \prod_{i=1}^d (t - m_i)\) where \(m_i\) are the exponents of \(W\) (e.g., \(A_d\): \(1,2,\dots,d\); \(B_d\): \(1,3,5,\dots,2d-1\); \(D_d\): \(1,3,\dots,2d-3, d-1\)). Tutte polynomial \(T_{\mathc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com