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arXiv · 1901.06893

Minimal tropical basis for Bergman fan of matroid

Yasuhito Nakajima
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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 $R_{10}$. 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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Yasuhito Nakajima. 2019-02-21. Minimal tropical basis for Bergman fan of matroid. https://arxiv.org/abs/1901.06893

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