Asymmetric tropical distances and power diagrams

TL;DR

This paper explores asymmetric tropical distances and their impact on Voronoi diagrams, revealing better-behaved structures and connections to power diagrams, with applications to rational lattices and Laurent modules.

Contribution

It introduces asymmetric tropical distances and demonstrates their advantages over symmetric versions, linking them to tropicalizations of power diagrams and extending their applications.

Findings

Asymmetric tropical Voronoi diagrams are better behaved than symmetric ones.

They can be viewed as tropicalizations of power diagrams over Puiseux series.

Applications include rational lattices and Laurent monomial modules.

Abstract

We investigate the Voronoi diagrams with respect to an asymmetric tropical distance function, also for infinite point sets. These turn out to be much better behaved than the tropical Voronoi diagrams arising from the standard tropical distance, which is symmetric. In particular, we show that the asymmetric tropical Voronoi diagrams may be seen as tropicalizations of power diagrams over fields of real Puiseux series. Our results are then applied to rational lattices and Laurent monomial modules.