Parallel Computation of tropical varieties, their positive part, and tropical Grassmannians

03/30/2020

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by   Dominik Bendle, et al.

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In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr_0(3,8), and show that it refines the 15-dimensional skeleton of the Dressian Dr(3,8) with the exception of 23 special cones for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional tropical cones which belong to the positive tropicalization. These algorithms exploit symmetries of the tropical variety even though the positive tropicalization need not be symmetric. We compute the maximal-dimensional cones of the positive Grassmannian TGr^+(3,8) and compare them to the cluster complex of the classical Grassmannian Gr(3,8).