We present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of symmetries using the workflow management system GPI-Space and the computer algebra system Singular. We determine the tropical Grassmannian TGr0(3,8). Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical S8-action in about 20 minutes. Relying on our result, we show that the Gröbner structure of TGr0(3,8) refines the 16-dimensional skeleton of the coarsest fan structure of the Dressian Dr(3,8), except for 23 orbits of special cones, for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional cones which belong to positive tropicalizations of algebraic varieties. We compute the positive Grassmannian TGr+(3,8) and compare it to the cluster complex of the classical Grassmannian Gr(3,8).

In this ISSAC 2021 Tutorial, we will give an introduction to the Singular/GPI-Space framework for massively parallel computations in computer algebra. Being based on the idea of separation of computation and coordination, this framework relies on the Petri net based workflow management system GPI-Space for the coordination layer, while the computer algebra system Singular is used as the frontend and computational backend. We will illustrate the use of this approach in examples taken from the context of algebraic geometry.