Differential Equations for Moving Hyperplane Arrangements

TL;DR

This paper studies the behavior of Mellin integrals of hyperplane products as hyperplanes move, aiming to develop a holonomic framework for these combinatorial correlators in hyperplane arrangements.

Contribution

It introduces a holonomic approach to analyze Mellin integrals of hyperplanes, providing a new method to encode these functions as holonomic functions.

Findings

Develops a holonomic annihilating $D$-ideal for hyperplane arrangements

Analyzes the behavior of combinatorial correlators under hyperplane movement

Provides a framework for encoding hyperplane integrals as holonomic functions

Abstract

We investigate Mellin integrals of products of hyperplanes, raised to an individual power each. We refer to the resulting functions as combinatorial correlators. We investigate their behavior when moving the hyperplanes individually. To encode these functions as holonomic functions in the constant terms of the hyperplanes, we aim to construct a holonomic annihilating $D$-ideal purely in terms of the hyperplane arrangement.