Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians

Bendle, Dominik; Böhm, Janko; Ren, Yue; Schröter, Benjamin

Abstract

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 . Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical -action in about 20 minutes. Relying on our result, we show that the Gröbner structure of refines the 16-dimensional skeleton of the coarsest fan structure of the Dressian , 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 and compare it to the cluster complex of the classical Grassmannian .

Citation

Bendle, D., Böhm, J., Ren, Y., & Schröter, B. (2023). Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians. Journal of Symbolic Computation, 120, https://doi.org/10.1016/j.jsc.2023.102224

• Journal Article Type: Article
• Acceptance Date: Apr 11, 2023
• Online Publication Date: Apr 26, 2023
• Publication Date: 2023
• Deposit Date: May 16, 2023
• Publicly Available Date: May 16, 2023
• Journal: Journal of Symbolic Computation
• Print ISSN: 0747-7171
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 120
• DOI: https://doi.org/10.1016/j.jsc.2023.102224