application: tropical

This application concentrates on tropical hypersurfaces and tropical polytopes. It provides the functionality for the computation of basic properties. Visualization and various constructions are possible.


imports from: common, graph
uses: polytope, topaz

Objects

User Methods of TropicalHypersurface

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    VISUAL () → Visual::TropicalHypersurface

    Visualizes the tropical hypersurface.

    Options
    option list: Visual::Polygons::decorations
    Returns
    Visual::TropicalHypersurface
  • Permutations of TropicalHypersurface

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    TropicalPolytope

    Tropical convex hull of finitely many points in the tropical (d-1)-torus, as introduced by Develin and Sturmfels. One construction is via the bounded subcomplex of an unbounded ordinary convex polyhedron.

    Properties of TropicalPolytope

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    PSEUDOVERTEX_GRAPH: graph::Graph<Undirected>

    Tropical polytopes have a natural description as ordinary polytopal complexes. This is the 1-skeleton of such a complex.

  • User Methods of TropicalPolytope

    Permutations of TropicalPolytope

    User Functions

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    cornered_hull (T) → TropicalPolytope

    Compute the cornered hull of a tropical polytope. Cf.

    M. Joswig, arXiv:0809.4694v2, Lemma 17.
    Parameters
    TropicalPolytope T
    Returns
    TropicalPolytope
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    dualize (points, generators) → Matrix

    Dualizes a point set with respect to the generators of a tropical polytope. The points are dualized with respect to the (rows of the) matrix of the generators. Cf.

    Develin & Sturmfels, Tropical Convexity, Lemma 22.
    Parameters
    Matrix points
    Matrix generators
    Returns
    Matrix
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    minkowski_sum (lambda, P, mu, Q) → TropicalPolytope

    Produces the tropical polytope lambda*P+mu*Q, where * and + are tropical scalar multiplication and tropical addition, respectively.

    Parameters
    Scalar lambda
    TropicalPolytope P
    Scalar mu
    TropicalPolytope Q
    Returns
    TropicalPolytope
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    cyclic (d, n) → TropicalPolytope

    Produces a tropical cyclic d-polytope with n vertices. Cf.

    Josephine Yu & Florian Block, arXiv: math.MG/0503279.
    Parameters
    int d
    the dimension
    int n
    the number of generators
    Returns
    TropicalPolytope
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    hypersimplex (k, d) → TropicalPolytope

    Produce the tropical hypersimplex Δ(k,d). Cf.

    M. Joswig math/0312068v3, Ex. 2.10.

    The value of k defaults to 1, yielding a tropical standard simplex.

    Parameters
    int k
    the number of -1 entries
    int d
    the dimension
    Returns
    TropicalPolytope
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    ch2d_3phases (n, Types, G) → Array<int>

    List the pseudovertices of a 2d tropical polytope on the boundary in counter-clockwise cyclic order.

    Parameters
    Int n
    the number of generators
    Array<Array<Set>> Types
    the types of the generators
    Graph G
    Returns
    Array<int>
    the pseudovertices on the boundary