Torische Ideale von Flusspolytopen

TL;DR

This thesis establishes degree bounds for generating sets and Gr"obner bases of toric ideals of flow polytopes, showing all are generated in degree three and providing bounds for Gr"obner bases with near-sharp examples.

Contribution

It proves degree bounds for toric ideals of flow polytopes and constructs examples demonstrating the near-sharpness of these bounds.

Findings

All toric ideals of flow polytopes are generated in degree three.

Smooth (3x4)-transportation polytopes are generated in degree two.

Reduced Gr"obner basis degree is at most mn/2, nearly sharp in examples.

Abstract

In dieser Diplomarbeit werden einige Gradschranken f\"ur Erzeugendensysteme und Gr\"obnerbasen von torischen Idealen von Flusspolytopen bewiesen. Alle torischen Ideale von Flusspolytopen sind im Grad 3 erzeugt. Glatte (3x4)-Transportpolytope sind sogar im Grad 2 erzeugt. Die reduzierte Gr\"obnerbasis eines beliebigen (m \times n)-Transportpolytops bez\"uglich einer beliebigen umgekehrt lexikographischen Termordnung hat h\"ochstens Grad mn/2. Wir konstruieren auch ein Beispiel, f\"ur das diese Schranke ann\"ahernd scharf ist. ----- In this Diplomarbeit (Master's thesis), we prove some degree bounds for generating sets and Gr\"obner bases of toric ideals of flow polytopes. All toric ideals of flow polytopes are generated in degree three. Smooth (3x4)-transportation polytopes are even generated in degree 2. The reduced Gr\"obner basis of an arbitrary (m \times n)-transportation polytope with respect to an arbitrary reverse lexikographic term order has at most degree mn/2. We also construct an example, for which this bound is almost sharp.