Pythagorean Hyperplane Arrangements: Combinatorics of Gain Genericity

TL;DR

This paper introduces a new concept of genericity for Pythagorean hyperplane arrangements, linking their combinatorics to an auxiliary arrangement, and explores initial applications and examples.

Contribution

It defines a novel genericity notion and constructs an auxiliary arrangement to analyze the combinatorics of Pythagorean hyperplane arrangements.

Findings

A new genericity concept for Pythagorean arrangements

Construction of an auxiliary arrangement linking to original arrangements

Initial applications and illustrative examples

Abstract

We study Pythagorean hyperplane arrangements, originally defined by Zaslavsky. In this first part of a series on such arrangements, we introduce a new notion of genericity for such arrangements. Using this notion we construct an auxiliary hyperplane arrangement whose combinatorics determines the combinatorics of all possible Pythagorean arrangements. We close with several applications and examples.