mixed_volume_classification

Sage code implementing the algorithms described in the paper Classification of triples of lattice polytopes with a given mixed volume as well as corresponding enumeration data. The folder python3 contains modified versions of the scripts, which are compatible with versions of SageMath running on top of Python3 (starting from version 9.0). This is joint work with Gennadiy Averkov and Ivan Soprunov.

Content:

Code:

• polytopes.sage, polytope_tuples.sage: basic operations on single lattice polytopes and tuples of lattice polytopes respectively
• full_dimensional_mvol_inductive.sage: classification of maximal full-dimensional triples of lattice polytopes of given mixed volume in dimension 3 (following Algorithm 5.4 of the paper)
• lower_dimensional_mvol_inductive.sage: classification of maximal irreducible triples of lattice polytopes of given mixed volume in dimension 3 (with at least one polytope not being full-dimensional, following Algorithm 6.3 of the paper)
• full_dimensional_mvol_polygons.sage: classification of maximal pairs of full-dimensional polygons of given mixed volume
• volume_classification.sage: classification of lattice polytopes of given volume in dimensions 2 and 3

Data:

This folder includes lists of maximal irreducible triples of mixed volume 1-4 as well as lists of pairs of full-dimensional polygons of mixed volume 1-10. Where it makes sense, the files containing the triples are split on the one hand by whether all of the polytopes in the triple are full-dimensional or not and on the other hand by whether the triple has the additional property of being R-maximal over the reals or not (see Definition 3.1) of the paper. Both triples and tuples are decoded as lists of the vertices of the respective polytopes and can e.g. be read into Sagemath using the function polytope_tuples_from_file from polytope_tuples.sage.

• dim_3_mv_[1-2].txt: maximal triples of full-dimensional lattice polytopes of mixed volume [1-2] in dimension 3
• dim_3_mv_[3-4]_rmax.txt: maximal triples of full-dimensional lattice polytopes of mixed volume [3-4] in dimension 3 that are R-maximal
• dim_3_mv_[3-4]_zmax.txt: maximal triples of full-dimensional lattice polytopes of mixed volume [3-4] in dimension 3 that are not R-maximal
• dim_3_mv_[2-4]_lower_d_rmax.txt: maximal irreducible triples of lattice polytopes of mixed volume [2-4] in dimension 3 that contain at least one 2-dimensional polytope and are R-maximal
• dim_3_mv_[2-4]_lower_d_zmax.txt: maximal irreducible triples of lattice polytopes of mixed volume [2-4] in dimension 3 that contain at least one 2-dimensional polytope and are not R-maximal
• dim_2_mv_[1-10].txt: maximal pairs of full-dimensional polygons with mixed volume [1-10]