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Tropical linear spaces

Introduction

A tropical linear space is a balanced polyhedral complex supported on a finite intersection of linear tropical hypersurfaces with all multiplicities one. It is dual to a matroid subdivision of a hypersimplex, and may arise as tropicalizations of linear ideals. For more on tropical linear spaces, see

Objects of type TropicalLinearSpace need to be embedded, abstract tropical linear spaces are currently not supported.

Constructors

In addition to converting from TropicalVariety, objects of type TropicalLinearSpace can be constructed from:

  1. Pluecker vectors over a tropical semiring: uses a low-level implementation in polymake
  2. Pluecker vectors over a field and a tropical semiring map: computes coordinatewise valuation and uses constructor (1.)
  3. matrices over a tropical semiring: computes tropical minors and uses constructor (1.)
  4. matrices over a field and a tropical semiring map.
    • if matrix over QQ and tropical semiring map is trivial, uses an implementation of Rincon's algorithm Rin13 in polymake
    • for general input, computes minors and uses constructor (2.)
tropical_linear_space

Properties

In addition to the properties inherited from TropicalVariety, objects of type TropicalLinearSpace have the following exclusive properties:

pluecker_indices(TropL::TropicalLinearSpace)
tropical_pluecker_vector(TropL::TropicalLinearSpace)
algebraic_pluecker_vector(TropL::TropicalLinearSpace)
tropical_semiring_map(TropL::TropicalLinearSpace)
tropical_matrix(TropL::TropicalLinearSpace)
algebraic_matrix(TropL::TropicalLinearSpace)
algebraic_ideal(TropL::TropicalLinearSpace)