Tropical varieties
Introduction
Tropial varieties (in OSCAR) are weighted polyhedral complexes. They may arise as tropicalizations of polynomial ideals or from operations on the more specialized types of tropical varieties, such as the stable intersection of tropical hypersurfaces. For more on the first, see
Objects of type TropicalVariety need to be embedded, abstract tropical varieties are currently not supported. The type TropicalVariety can be thought of as supertype of TropicalHypersurface, TropicalCurve, and TropicalLinearSpace in the sense that the latter three inherit all properties and features of the former.
Constructor
Objects of type TropicalVariety can be constructed as follows:
tropical_variety — Functiontropical_variety(Sigma::PolyhedralComplex, mult, minOrMax::Union{typeof(min),typeof(max)}=min)Return the TropicalVariety whose polyhedral complex is Sigma with multiplicities mult and convention minOrMax. Here, mult is optional can be specified as a Vector{ZZRingElem} which represents a list of multiplicities on the maximal polyhedra in the order of maximal_polyhedra(Sigma). If mult is unspecified, then all multiplicities are set to one.
Examples
julia> Sigma = polyhedral_complex(IncidenceMatrix([[1],[2]]), [[0],[1]])
Polyhedral complex in ambient dimension 1
julia> tropical_variety(Sigma)
Min tropical variety
julia> mult = ones(ZZRingElem, n_maximal_polyhedra(Sigma))
2-element Vector{ZZRingElem}:
1
1
julia> tropical_variety(Sigma,mult,min)
Min tropical variety
julia> mult = ZZ.([1,2])
2-element Vector{ZZRingElem}:
1
2
julia> tropical_variety(Sigma,mult,max)
Max tropical variety
Properties
Objects of type TropicalVariety (and TropicalHypersurface, TropicalCurve, TropicalLinearSpace) have the following properties:
polyhedral_complex — Methodpolyhedral_complex(TropV::TropicalVariety)Return the polyhedral complex of a tropical variety.
ambient_dim — Methodambient_dim(TropV::TropicalVariety)Return the ambient dimension of TropV. Requires TropV to be embedded.
codim — Methodcodim(TropV::TropicalVariety)Return the codimension of TropV. Requires TropV to be embedded.
dim — Methoddim(TropV::TropicalVariety)Return the dimension of TropV.
f_vector — Methodf_vector(TropV::TropicalVariety)Return the f-Vector of TropV.
lineality_dim — Methodlineality_dim(TropV::TropicalVariety)Return the dimension of the lineality space of TropV.
lineality_space — Methodlineality_space(TropV::TropicalVariety)Return the lineality space of TropV.
maximal_polyhedra — Methodmaximal_polyhedra(TropV::TropicalVariety)Return the maximal polyhedra of TropV.
maximal_polyhedra_and_multiplicities — Methodmaximal_polyhedra_and_multiplicities(TropV::TropicalVariety)Return the maximal polyhedra of TropV.
minimal_faces — Methodminimal_faces(TropV::TropicalVariety)Return the minimal faces of TropV.
multiplicities — Methodmultiplicities(TropV::TropicalVariety)Return the multiplicities of TropV.
n_maximal_polyhedra — Methodn_maximal_polyhedra(TropV::TropicalVariety)Return the number of maximal polyhedra of TropV.
n_polyhedra — Methodn_polyhedra(TropV::TropicalVariety)Return the number of polyhedra of TropV.
n_vertices — Methodn_vertices(TropV::TropicalVariety)Return the number of vertices of TropV.
is_pure — Methodis_pure(TropV::TropicalVariety)Return true if TropV is pure as a polyhedral complex, false otherwise.
is_simplicial — Methodis_simplicial(TropV::TropicalVariety)Return true if TropV is simplicial as a polyhedral complex, false otherwise.
rays — Methodrays(TropV::TropicalVariety)Return the rays of TropV.
rays_modulo_lineality — Methodrays_modulo_lineality(TropV::TropicalVariety)Return the rays modulo the lineality space of TropV.
vertices_and_rays — Methodvertices_and_rays(TropV::TropicalVariety)Return the vertices and rays of TropV.
vertices — Methodvertices(TropV::TropicalVariety)Return the vertices of TropV.
visualize — Methodvisualize(TropV::TropicalVariety)Visualize TropV.