TY  - JOUR
A1  - Bendle, Dominik
A1  - Böhm, Janko
A1  - Ren, Yue
A1  - Schröter, Benjamin Frederik
T1  - Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians
T2  - Journal of symbolic computation
N2  - 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).
KW  - Tropical geometry
KW  - Tropical varieties
KW  - Tropical Grassmannians
Y1  - 2023
UR  - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/75896
UR  - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-758964
SN  - 0747-7171
VL  - 120.2024
IS  - 102224
PB  - Elsevier
CY  - Amsterdam
ER  -