Minimal tropical basis for Bergman fan of matroid
Abstract
The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuits set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and R10. We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.
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