EPH-classifications in Geometry, Algebra, Analysis and Arithmetic

TL;DR

This paper investigates the widespread EPH-classification across various mathematical fields, examining its justification and whether it reflects a shared underlying phenomenon among elliptic, parabolic, and hyperbolic cases.

Contribution

The paper provides a comprehensive analysis of the EPH-classification's applicability and explores its conceptual unity across geometry, algebra, analysis, and arithmetic.

Findings

EPH-classification appears in multiple mathematical areas.

The justification for EPH-classification varies across contexts.

Shared mathematical structures may underlie the EPH-trichotomy.

Abstract

Trichotomy of Elliptic-Parabolic-Hyperbolic appears in many different areas of mathematics. All of these are named after the very first example of trichotomy, which is formed by ellipses, parabolas, and hyperbolas as conic sections. We try to understand if these classifications are justified and if similar mathematical phenomena is shared among different cases EPH-classification is used.