Singular/GPI-Space Tropicalization Project

Contains the raw data (compressed ~100MB; uncompressed ~7GB) produced by the parallel computation with the Singular/GPIspace application tropicalgspc. Each file in the tar file encodes one of the 14763 Gröbner cone representatives with respect to the S₈-symmetry action and includes a standard basis of the initial ideal inside the polynomial ring, and inequalities and equations of the polyhedral cone. Note that both variables and coordinates are always ordered lexicographically while the termorder depends on the cone.

The singular script singular_construct_TGr38GroebnerBasis.sing produces a list with all 14763 initial ideals and executing the singular script singular_construct_TGr38GroebnerCones.sing results in a list of all the polyhedral cones.

We also provide the data in a human-readable serialization format, see singular_data_TGR38Groebner_ideals_text.zip and singular_data_TGR38Groebner_cones_text.zip. The first contains for each ideal a list of polynomials generating the ideal, the second contains for each maximal cone of the tropical variety a description of the cone using homogeneous inequalities and equations, both of which are encoded as integer vectors.

This file contains a polymake array (Array<Array<Int>>) listing all 40320 permutations specifiying the S₈-action on the 56 coordinates. The data can be loaded into polymake via the command

load_data("polymake_data_Group38.data");

This file contains a polymake matrix (Matrix<Rational>) whose rows contain all 95 representatives of the rays of the tropical Grassmannian(3,8) with the Gröbner structure. This file can also be loaded via

load_data("polymake_data_TGr38GroebnerRays.data");

This file contains a polymake array (Array<Set<String>>) of 14763 sets that encode the maximal dimensional cones. The sets contain strings with pairs of indices of rays and permutations. The cones can be extracted with the polymake script read_cones.pl.

The first 12 rows of the file list representatives of the rays of the Dressian modulo the group action on the Dressian given by the permutations as specified above in the file polymake_data_Group38.data. Note, that the remaining rows contain the remaining rays of the tropical Grassmannian(3,8) with the Gröbner structure and are not relevant for the Plücker structure (see above). Note, that the full polyhedral structure of the Dressian has been computed by Herrmann, Speyer and Joswig, see the reference given in the above mentioned paper.

This file contains a polymake array of 4766 sets that encode the maximal dimensional cones encodes as described above. The cones can be extracted via the polymake script read_cones.pl.

This Singular script constructs a list of 4766 integer vectors (of type intvec). Each is a relative interior point of one of a total of 4766 representatives of  the16-dimensional cones of the tropical Grassmannian with the coarsest structure. See the file for comments with more details.

The data can be loaded into Singular via the command:
 <"singular_construct_TGr38Pluecker.sing";

Constructs a Singular list of 4766 lists. The i-th list contains a (possibly empty) list of permutations which map the i-th relative interior point into the positive tropicalization. Here, a permutation is encoded as vector (of type intvec).  See the file for comments with more details.

The data can be loaded into Singular via the command:
 <"singular_construct_PosTGr38Pluecker.sing";

This file contains the positive part of the tropical Grassmannian as a subfan of the secondary fan of the hypersimplex provided as a polymake 'fan' object which can be loaded in polymake by executing the command

load("polymake_data_PosTGr38Pluecker.fan");

This file contains a refinement of the above listed subfan of the Plücker fan. The refined fan equips the positive part of the tropical Grassmannian with the structure of the cluster complex. This data can be loaded in polymake via

load("polymake_data_PosTGr38Cluster.fan");

This is a polymake script that compares the cluster complex as provided by sage (which is hard coded in the script) to the fan as provided in the above file, verifying that the two fans coincide.

A collection of procedures that were used in the computation (the scripts will be merged into the main Singular repository in form of a library).

Herrmann, Joswig, Speyer: Dressians, Tropical Grassmannians, and Their Rays. Forum Math. 26, No. 6, 1853-1881 (2014) (arXiv:1112.1278; data of the Dressian can be found at http://svenherrmann.net/DR38/dr38.html).