The Gr\"obner stratification of a tropical variety

TL;DR

This paper investigates the structure of Gr"obner stratification in tropical varieties, showing it can be finer than polyhedral decompositions and is determined by the underlying tropical variety in certain cases.

Contribution

It introduces the concept of Gr"obner stratification as a refinement of polyhedral decomposition, independent of compactification, and characterizes it for locally matroidal tropical varieties.

Findings

Gr"obner stratification can be strictly finer than polyhedral decomposition.

The stratification is independent of compactification choices.

For locally matroidal tropical varieties, the stratification is determined by the tropical variety.

Abstract

Each Gr\"obner stratum of a tropical variety is a connected set of points, all of which induce the same initial subscheme. The Gr\"obner stratification is a coarsening of the decomposition into Gr\"obner polyhedra, and has the advantage that it does not depend on a choice of compactification. We give an example of a curve over a field with non-trivial valuation whose Gr\"obner stratification is strictly finer than the coarsest polyhedral decomposition of the tropical variety. We also show that the Gr\"obner stratification of a locally matroidal tropical variety is completely determined by the underlying tropical variety.