Polyhedral Geometry in OSCAR

TL;DR

This paper introduces OSCAR, a new computer algebra system that integrates tools for polyhedral geometry, providing accessible computation methods and case studies for advanced polyhedral analysis.

Contribution

It presents an introduction to polyhedral geometry computations within OSCAR, combining multiple algebraic systems and demonstrating practical applications through detailed case studies.

Findings

Methods for computing convex hulls and linear programs in OSCAR

Analysis of face numbers of random polytopes

Properties of Gelfand-Tsetlin and secondary polytopes

Abstract

OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number theory). Assuming little familiarity with the subject, we give an introduction to computations in polyhedral geometry using OSCAR, as a chapter of the upcoming OSCAR book. In particular, we define polytopes, polyhedra, and polyhedral fans, and we give a brief overview about computing convex hulls and solving linear programs. Three detailed case studies are left for experts in polyhedral geometry. These are concerned with face numbers of random polytopes, constructions and properties of Gelfand-Tsetlin polytopes, and secondary polytopes.