The Gröbner stratification of a tropical variety

Each Gröbner stratum of a tropical variety is a connected set of points, all of which induce the same initial subscheme. The Gröbner stratification is a coarsening of the decomposition into Gröbner 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 Gröbner stratification is strictly finer than the coarsest polyhedral decomposition of the tropical variety. We also show that the Gröbner stratification of a locally matroidal tropical variety is completely determined by the underlying tropical variety.

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