Properties of Cycle

•  

  BALANCED_FACES: common::Vector<Bool>
  

  A vector whose entries correspond to the rows of CODIMENSION_ONE_POLYTOPES. The i-th entry is true, if and only if the complex is balanced at that face

•  

  IS_BALANCED: common::Bool
  

  Whether the cycle is balanced. As many functions in a-tint can deal with non-balanced complexes, we include this as a property.

•  

  IS_IRREDUCIBLE: common::Bool
  

  Whether this complex is irreducible.

•  

  LATTICE_BASES: common::IncidenceMatrix<NonSymmetric>
  

  This incidence matrix gives a lattice basis for each maximal polytope. More precisely it gives a lattice basis whose span contains the lattice of the maximal polytope. Row i corresponds to cone i and gives lattice generator indices referring to LATTICE_GENERATORS. If this property is computed via rules, it does indeed give a lattice basis for the cone lattice, but when it is computed during an operation like refinement or divisor it will in general be larger. If this property exists, lattice normals might be computed faster.

•  

  LATTICE_GENERATORS: common::Matrix<Integer, NonSymmetric>
  

  This is an irredundant list of all lattice generators of all maximal polyhedra. If this property exists, lattice normals might be computed faster

•  

  LATTICE_NORMAL_FCT_VECTOR: common::Map<Pair<Int, Int>, Vector<Rational>>
  

  For each lattice normal vector, this gives a vector of length (number of rays) + (lineality dim.), such that if a rational function is given by values on the rays and lin space generators, the value of the corresponding normal LATTICE_NORMALS->{i}->{j} can be computed by multiplying the function value vector with the vector LATTICE_NORMAL_FCT_VECTOR->{i}->{j}. This is done in the following way: We use the generating system (and indices refer to SEPARATED_VERTICES) <(r_i-r_0)_i>0, s_j, l_k>, where r_0 is the ray of the maximal cone with the lowest index in SEPARATED_VERTICES, such that it fulfills x0 = 1, r_i are the remaining rays with x0 = 1, ordered according to their index in SEPARATED_VERTICES, s_j are the rays of the cone with x0 = 0 and l_k are the lineality space generators. We will then store the coefficients a_i of (r_i - r_0) at the index of r_i, then - sum(a_i) at the index of r_0 and the remaining coefficients at the appropriate places. In particular, the value of a lattice normal under a rational function can be computed simply by taking the scalar product of RAY_VALUES | LIN_VALUES with this FCT_VECTOR

•  

  LATTICE_NORMAL_SUM: common::Matrix<Rational, NonSymmetric>
  

  Rows of this matrix correspond to CODIMENSION_ONE_POLYTOPES, and each row contains the weighted sum: sum_{cone > codim-1-face}( weight(cone) * LATTICE_NORMALS->{codim-1-face}->{cone})

•  

  LATTICE_NORMAL_SUM_FCT_VECTOR: common::Matrix<Rational, NonSymmetric>
  

  Rows of this matrix correspond to SEPARATED_CODIMENSION_ONE_POLYTOPES and each row contains a function vector for the corresponding row of LATTICE_NORMAL_SUM. This function vector is computed in the same way as described under LATTICE_NORMAL_FCT_VECTOR. Note that for any codim-1-faces at which the complex is not balanced, the corresponding row is a zero row. If a face is balanced can be checked under BALANCED_FACES.

•  

  WEIGHT_CONE: polytope::Cone<Rational>
  

  The intersection of WEIGHT_SPACE with the positive orthant.

•  

  WEIGHT_SPACE: common::Matrix<Rational, NonSymmetric>
  

  A Z-basis (as rows) for the space of weight distributions on this tropical cycle making it balanced (i.e. this cycle is irreducible, if and only if WEIGHT_SPACE has only one row and the gcd of WEIGHTS is 1.

•  

  WEIGHT_SYSTEM: common::Matrix<Rational, NonSymmetric>
  

  The dual of WEIGHT_SPACE.

•  

  Defining morphisms and functions

  These properties are used to define morphisms or rational functions on a Cycle.

  •  

    SEPARATED_CODIMENSION_ONE_POLYTOPES: common::IncidenceMatrix<NonSymmetric>
    

    An incidence matrix describing which codimension one polytope in the complex is generated by which vertices. Each row corresponds to a codimension one polytope (More precisely, the i-th element represents the same codim 1 polytope as the i-th element of CODIMENSION_ONE_POLYTOPES). The indices in a row refer to rows of SEPARATED_VERTICES.

  •  

    SEPARATED_CONVERSION_VECTOR: common::Vector<Int>
    

    A vector with an entry for each row in SEPARATED_VERTICES. More precisely, the i-th entry gives the row index of the ray in VERTICES that is equal to the i-th row of SEPARATED_VERTICES.

  •  

    SEPARATED_MAXIMAL_POLYTOPES: common::IncidenceMatrix<NonSymmetric>
    

    An incidence matrix describing which maximal polytope in the complex us generated by which rays. Each row corresponds to a maximal polytope (More precisely, the i-th element represents the same maximal polytope as the i-th element of MAXIMAL_POLYTOPES). The indices in a row refer to rows of SEPARATED_VERTICES, i.e. the maximal polytope described by the i-th element is generated by the vertices corresponding to these row indices.

  •  

    SEPARATED_VERTICES: common::Matrix<Rational, NonSymmetric>
    

    This is a matrix of vertices of the complex. More precisely, each ray r from VERTICES occurs as a row in this matrix... - once, if r_0 = 1 - k times, if r_0 = 0 and k is the number of equivalence classes of maximal cones containing r with respect to the following relation: Two maximal cones m, m' containing r are equivalent, if they are equal or there exists a sequence of maximal cones m = m_1,...m_r = m', such that r is contained in each m_i and each intersection m_i cap m_i+1 contains at least one ray s with s_0 = 1. The reason for this is that, when for example specifying a piecewise affine linear function on a polyhedral complex, the same far ray with x0 = 0 might be assigned two different values, if it is contained in two "non-connected" maximal cones (where connectedness is to be understood as described above). If there is a LOCAL_RESTRICTION the above equivalence relation is changed in such a way that the affine ray s with s_0 = 1 that must be contained in the intersection of two subsequent cones must be a compatible ray

•  

  Intersection theory

  These are general properties related to intersection theory.

  •  

    DEGREE: common::Integer
    

    The degree of the tropical variety, i.e. the weight of the intersection product with a uniform tropical linear space of complementary dimension.

•  

  Local computations

  These properties are used for doing computations locally around a specified part of a Cycle. ----- These +++ deal with the creation and modification of cycles with nontrivial LOCAL_RESTRICTION.

  •  

    LOCAL_RESTRICTION: common::IncidenceMatrix<NonSymmetric>
    

    This contains a list of sets of ray indices (referring to VERTICES). All of these sets should describe polyhedra of the polyhedral complex (though not necessarily maximal ones). A polyhedron is now called compatible with this property, if it contains one of these polyhedra If this list is not empty, all computations will be done only on (or around) compatible cones. The documentation of each property will explain in what way this restriction is enforced. If this list is empty or not defined, there is no restriction. Careful: The implementation assumes that ALL maximal cones are compatible. If in doubt, you can create a complex with a local restriction from a given complex by using one of the "local_..." creation methods This list is assumed to be irredundant, i.e. there are no doubles (though this should not break anything, it is simply less efficient). It is, however, possible that one element is a subset of another.

User Methods of Cycle

•  

  Basic polyhedral operations

  These methods provide basic functionality related to polyhedral geometry, but not necessarily to tropical geometry

  •  

    is_fan (allow_translations) → Bool

    
    

    Checks whether this polyhedral structure is a fan, i.e. has only a single vertex at the origin.

    Parameters

    Bool

    allow_translations

    . Optional and false by default. If true, a shifted fan is also accepted.

    Returns

    Bool

    .

•  

  Local computations

  These methods are used for doing computations locally around a specified part of a Cycle. ----- These +++ deal with the creation and modification of cycles with nontrivial LOCAL_RESTRICTION.

  •  

    delocalize () → Cycle

    
    

    Returns the cycle without its LOCAL_RESTRICTION (Note that only the defining properties are kept. All derived information is lost).

    Returns

    Cycle

•  

  Visualization

  These methods are for visualization.

  •  

    BB_VISUAL ()

    
    

    Same as VISUAL. Kept for backwards compatibility.
  •  

    bounding_box (dist, chart)

    
    

    Takes a chart and a positive Rational as input and computes the relative bounding box of the cycle, i.e. it takes the coordinate-wise minimum and maximum over the coordinates of the nonfar vertices and adds/subtracts the given number. This is returned as a 2xdim matrix.

    Parameters

    Rational	dist	. Optional, 1 by default.
    Int	chart	. The chart to be used fot he computation.

  •  

    VISUAL ()

    
    

    Displays a (possibly weighted) polyhedral complex by intersecting it with a bounding box. This bounding box is either defined by the vertices of the complex and the option "BoundingDistance" or explicitly given by "BoundingBox" and by setting "BoundingMode" to "absolute" @options

    Options

    Int	Chart	Which affine chart to visualize, i.e. which coordinate to shift to 0. This is 0 by default.
    String	WeightLabels	If "hidden", no weight labels are displayed. Not hidden by default.
    String	CoordLabels	If "show", coordinate labels are displayed at vertices. Hidden by default.
    String	BoundingMode	Can be "relative" (intersects with the bounding box returned by the method boundingBox(BoundingDistance)) or "absolute" (intersects with the given BoundingBox) or "cube" (essentially the same as "relative", but the bounding box is always a cube). "cube" by default.
    Rational	BoundingDistance	The distance parameter for relative bounding mode
    Matrix<Rational>	BoundingBox	The bounding parameter for absolute bounding mode

Properties of LinesInCubic

•  

  Counts of lines

  These count lines or families of lines in the cubic.# @category Defining properties

  •  

    N_FAMILIES: common::Int
    

    The total number of families in LIST_FAMILY_FIXED_EDGE, LIST_FAMILY_FIXED_VERTEX, LIST_FAMILY_MOVING_EDGE and LIST_FAMILY_MOVING_VERTEX

  •  

    N_ISOLATED: common::Int
    

    The total number of elements in LIST_ISOLATED_EDGE and LIST_ISOLATED_NO_EDGE

•  

  Defining properties

  The polynomial and the corresponding cubic.

  •  

    CUBIC: Cycle
    

    The surface containing the lines

  •  

    POLYNOMIAL: common::Polynomial
    

    The homogeneous tropical polynomial defining the surface

•  

  Lists of lines

  These contain lists of certain (families of) lines.

  •  

    LIST_FAMILY_FIXED_EDGE: Cycle as LinesInCubic::LIST_FAMILY_FIXED_EDGE
    

    A list of all families of lines with (part of) the bounded edge fixed

  •  

    LIST_FAMILY_FIXED_VERTEX: Cycle as LinesInCubic::LIST_FAMILY_FIXED_VERTEX
    

    A list of all families of lines without a bounded edge at a fixed vertex

  •  

    LIST_FAMILY_MOVING_EDGE: Cycle as LinesInCubic::LIST_FAMILY_MOVING_EDGE
    

    A list of all families of lines with the bounded edge moving in transversal direction

  •  

    LIST_FAMILY_MOVING_VERTEX: Cycle as LinesInCubic::LIST_FAMILY_MOVING_VERTEX
    

    A list of all families of lines without a bounded edge and a moving vertex.

  •  

    LIST_ISOLATED_EDGE: Cycle as LinesInCubic::LIST_ISOLATED_EDGE
    

    A list of all isolated lines with bounded edge

  •  

    LIST_ISOLATED_NO_EDGE: Cycle as LinesInCubic::LIST_ISOLATED_NO_EDGE
    

    A list of all isolated lines without a bounded edge

User Methods of LinesInCubic

•  

  Lists of lines

  These contain lists of certain (families of) lines.

  •  

    all_families () → Cycle<Addition>

    
    

    UNDOCUMENTED

    Returns

    Cycle<Addition>

    A perl array containing all families

  •  

    all_isolated () → Cycle<Addition>

    
    

    UNDOCUMENTED

    Returns

    Cycle<Addition>

    A perl array containing all isolated solutions

  •  

    array_family_fixed_edge () → Cycle<Addition>

    
    

    UNDOCUMENTED

    Returns

    Cycle<Addition>

    A perl array version of LIST_FAMILY_FIXED_EDGE

  •  

    array_family_fixed_vertex () → Cycle<Addition>

    
    

    UNDOCUMENTED

    Returns

    Cycle<Addition>

    A perl array version of LIST_FAMILY_FIXED_VERTEX

  •  

    array_family_moving_edge () → Cycle<Addition>

    
    

    UNDOCUMENTED

    Returns

    Cycle<Addition>

    A perl array version of LIST_FAMILY_MOVING_EDGE

  •  

    array_family_moving_vertex () → Cycle<Addition>

    
    

    UNDOCUMENTED

    Returns

    Cycle<Addition>

    A perl array version of LIST_FAMILY_MOVING_VERTEX

  •  

    array_isolated_edge () → Cycle<Addition>

    
    

    UNDOCUMENTED

    Returns

    Cycle<Addition>

    A perl array version of LIST_ISOLATED_EDGE

  •  

    array_isolated_no_edge () → Cycle<Addition>

    
    

    UNDOCUMENTED

    Returns

    Cycle<Addition>

    A perl array version of LIST_ISOLATED_NO_EDGE

Properties of MatroidRingCycle

•  

  Matroid data

  These properties are concerned with the underlying matroid combinatorics of a MatroidRingCycle.

  •  

    NESTED_COEFFICIENTS: common::Array<Int>
    

    These are the linear coefficients of this cycle in its nested matroids representation. They correspond to the entries in NESTED_PRESENTATIONS.

  •  

    NESTED_PRESENTATIONS: common::Array<IncidenceMatrix<NonSymmetric>>
    

    Every matroid (and hence every element of the matroid ring) has a presentation as a linear combination of nested matroids (which form a basis). Each entry in this list represents a nested matroid whose coefficient is not 0 in the presentation of this cycle. A nested matroid is represented by its maximal transversal presentation. You can use the method nested_matroids to retrieve them as actual matroid::Matroid objects. Note that the implementation will assume that the nested presentations are each given in such a way that the sets are ordered from smallest to largest. It is not recommended to specify this property directly. Rather use the constructor to convert a matroid into a ring cycle, then use the overloaded operators to construct linear combinations.

  •  

    N_ELEMENTS: common::Int
    

    Describes the ground set of the matroid ring in which this cycle lives. Same as PROJECTIVE_AMBIENT_DIM+1.

  •  

    N_NESTED_SUMMANDS: common::Int
    

    The number of entries in NESTED_PRESENTATIONS (or NESTED_COEFFICIENTS).

  •  

    RANK: common::Int
    

    Describes the ranks of the matroids, of which this cycle is a linear combination. Same as PROJECTIVE_DIM+1

User Methods of MatroidRingCycle

•  

  Matroid data

  These methods are concerned with the underlying matroid combinatorics of a MatroidRingCycle.

  •  

    nested_matroids ()

    
    

    This returns the nested matroids represented by NESTED_PRESENTATIONS as a list of matroid::Matroid objects.

    
    

    Example:

Properties of Morphism

•  

  Defining morphisms and functions

  These properties are used to define morphisms or rational functions on a Cycle.

  •  

    DOMAIN: Cycle
    

    This property describes the domain of the morphism. I.e. the morphism is defined on this complex and is locally affine integral linear.

  •  

    IS_GLOBALLY_AFFINE_LINEAR: common::Bool
    

    This is TRUE, if the morphism is defined on the full projective torus by a MATRIX and a TRANSLATE The rules do not actually check for completeness of the DOMAIN. This property will be set to TRUE, if the morphism is only defined by MATRIX and TRANSLATE, otherwise it is false (or you can set it upon creation).

  •  

    LINEALITY_VALUES: common::Matrix<Rational, NonSymmetric>
    

    The vector in row i describes the function value (slope) of DOMAIN->LINEALITY_SPACE->row(i)

  •  

    MATRIX: common::Matrix<Rational, NonSymmetric>
    

    If the morphism is a global affine linear map x |-> Ax+v, then this contains the matrix A. Note that this must be well-defined on tropical projective coordinates, so the sum of the columns of A must be a multiple of the (1,..,1)-vector. If TRANSLATE is set, but this property is not set, then it is the identity by default.

  •  

    TRANSLATE: common::Vector<Rational>
    

    If the morphism is a global affine linear map x |-> Ax+v, then this contains the translation vector v. If MATRIX is set, but this property is not set, then it is the zero vector by default.

  •  

    VERTEX_VALUES: common::Matrix<Rational, NonSymmetric>
    

    The vector at row i describes the function value of vertex DOMAIN->SEPARATED_VERTICES->row(i). (In tropical homogenous coordinates, but without leading coordinate). More precisely, if the corresponding vertex is not a far ray, it describes its function value. If it is a directional ray, it describes the slope on that ray.

User Methods of Morphism

•  

  Affine and projective coordinates

  These methods deal with affine and projective coordinates, conversion between those and properties like dimension that change in projective space.

  •  

    affine_representation (domain_chart, target_chart) → Pair<Matrix<Rational>, Vector<Rational>>

    
    

    Computes the representation of a morphism (given by MATRIX and TRANSLATE) on tropical affine coordinates.

    Parameters

    Int	domain_chart	Which coordinate index of the homogenized domain is shifted to zero to identify it with the domain of the affine function. 0 by default.
    Int	target_chart	Which coordinate of the homogenized target space is shifted to zero to identify it with the target of the affine function. 0 by default.

    Returns

    Pair<Matrix<Rational>, Vector<Rational>>

    A matrix and a translate in affine coordinates.

•  

  Morphisms

  These are general methods that deal with morphisms and their arithmetic.

  •  

    after (g) → Morphism

    
    

    Computes the composition of another morphism g and this morphism. This morphism comes after g.

    Parameters

    Morphism

    g

    Returns

    Morphism

    this after g

  •  

    before (g) → Morphism

    
    

    Computes the composition of this morphism and another morphism g. This morphism comes before g.

    Parameters

    Morphism

    g

    Returns

    Morphism

    g after f

  •  

    restrict (Some) → Morphism

    
    

    Computes the restriction of the morphism to a cycle. The cycle need not be contained in the DOMAIN of the morphism, the restriction will be computed on the intersection.

    Parameters

    Cycle

    Some

    cycle living in the same ambient space as the DOMAIN

    Returns

    Morphism

    The restriction of the morphism to the cycle (or its intersection with DOMAIN.

•  

  Visualization

  These methods are for visualization.

  •  

    VISUAL ()

    
    

    Visualizes the domain of the morphism. Works exactly as VISUAL of WeightedComplex, but has additional option @options

    Options

    String

    FunctionLabels

    If set to "show", textual function representations are diplayed on cones. False by default

Properties of RationalCurve

•  

  Combinatorics

  These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

  •  

    COEFFS: common::Vector<Rational>
    

    A list of positive rational coefficients. The list should have the same length as SETS and contain only entries > 0. The i-th entry then gives the length of the bounded edge defined by the i-th partition. If you're not sure if all your coefficients are > 0, use INPUT_SETS and INPUT_COEFFS instead. Note that the zero curve (i.e. no bounded edges, only leaves) is represented by one empty set with corresponding lenghth 0.

  •  

    GRAPH: graph::Graph<Undirected>
    

    Contains the abstract graph (non-metric) corresponding to the curve. All unbounded leaves are modelled as bounded edges. The vertices at the ends of the "leaves" are always the first N_LEAVES vertices.

  •  

    GRAPH_EDGE_LENGTHS: common::Vector<Rational>
    

    Contains the lengths of the edges of GRAPH that represent bounded edges of the curve. The coefficients appear in the order that the corr. edges appear in EDGES.

  •  

    NODES_BY_LEAVES: common::IncidenceMatrix<NonSymmetric>
    

    This incidence matrix gives a list of the vertices of the curve Each row corresponds to a vertex and contains as a set the [[LEAVES] that are attached to that vertex (again, counting from 1!)

  •  

    NODES_BY_SETS: common::IncidenceMatrix<NonSymmetric>
    

    This incidence matrix gives a list of the vertices of the curve Each row corresponds to a vertex and contains as a set the row indices of the SETS that correspond to edges attached to that vertex

  •  

    NODE_DEGREES: common::Vector<Int>
    

    This gives a list of the vertices of the curve in terms of their valences They appear in the same order as in NODES_BY_LEAVES or NODES_BY_SETS

  •  

    N_LEAVES: common::Int
    

    The number of leaves of the rational curve.

  •  

    SETS: common::IncidenceMatrix<NonSymmetric>
    

    A list of partitions of [n] that define the tree of the curve: For each bounded edge we have the corresponding partition of the n leaves. These should be irredundant. If you want to input a possibly redundant list, use INPUT_SETS and INPUT_COEFFS instead. The number of marked leaves should always be given by N_LEAVES. The sets are subsets of {1,...,n} (NOT {0,..,n-1}!) Note that the zero curve (i.e. no bounded edges, only leaves) is represented by one empty set with corresponding lenghth 0.

•  

  Input property

  These properties are for input only. They allow redundant information.

  •  

    INPUT_COEFFS: common::Vector<Rational>
    

    Same as COEFFS, except that entries may be <=0. This should have the same length as INPUT_SETS.

  •  

    INPUT_SETS: common::IncidenceMatrix<NonSymmetric>
    

    Same as SETS, except that sets may appear several times.

  •  

    INPUT_STRING: common::String
    

    This property can also be used to define a rational curve: A linear combination of partitions is given as a string, using the following syntax: A partition is given as a subset of {1,..,n} and written as a comma-separated list of leaf indices in round brackets, e.g. "(1,2,5)" A linear combination can be created using rational numbers, "+","+" and "-" in the obvious way, e.g. "2*(1,2,5) + 1*(3,4,7) - 2(1,2) (The "*" is optional) Of course, each set should contain at least two elements. If you don't specify N_LEAVES, it is set to be the largest leaf index occuring in the sets. Partitions needn't be irredundant and coefficients can be any rational number. If the resulting element is not in the moduli space, an error is thrown.

User Methods of RationalCurve

•  

  Conversion

  These deal with converting the representation of a rational curve between metric vector and matroid fan coordinates.

  •  

    metric_vector ()

    
    

    Returns the (n over 2) metric vector of the rational n-marked curve
•  

  Visualization

  These methods are for visualization.

  •  

    VISUAL ()

    
    

    Visualizes a RationalCurve object. This visualization uses the VISUAL method of its GRAPH, so it accepts all the options of Visual::Graph::decorations. In addition it has another option @options

    Options

    String

    LengthLabels

    If "hidden", the edges are not labelled with their lengths. Any other text is ignored. Not set to "hidden" by default.

Properties of RationalFunction

•  

  Defining morphisms and functions

  These properties are used to define morphisms or rational functions on a Cycle.

  •  

    DENOMINATOR: common::Polynomial
    

    When representing the function as a quotient of tropical polynomials, this is the denominator. Should be a homogeneous polynomial of the same degree as NUMERATOR.

  •  

    DOMAIN: Cycle
    

    This property describes the affine linearity domains of the function. I.e. the function is affine integral linear on each maximal polytope of DOMAIN.

  •  

    IS_GLOBALLY_DEFINED: common::Bool
    

    This is TRUE, if the function is defined on the full projective torus by a NUMERATOR and a DENOMINATOR. The rules do not actually check for completeness of the DOMAIN. This property will be set to true, if the function is created only via NUMERATOR and DENOMINATOR. Otherwise it will be set to FALSE (or you can set it manually upon creation).

  •  

    LINEALITY_VALUES: common::Vector<Rational>
    

    The value at index i describes the function value of DOMAIN->LINEALITY_SPACE->row(i)

  •  

    NUMERATOR: common::Polynomial
    

    When representing the function as a quotient of tropical polynomials, this is the numerator. Should be a homogeneous polynomial of the same degree as DENOMINATOR.

  •  

    POWER: common::Int
    

    This is an internally used property that should not actually be set by the user. When creating a rational function with the ^-operator, this property is set to the exponent. The semantics is that when computing a divisor, this function should be applied so many times The usual application of this is a call to divisor($X, $f^4) or something similar. Warning: This property is not stored if the RationalFunction object is saved. Nor should be assumed to be preserved during any kind of arithmetic or restricting operation.

  •  

    VERTEX_VALUES: common::Vector<Rational>
    

    The value at index i describes the function value at DOMAIN->SEPARATED_VERTICES->row(i). More precisely, if the corresponding vertex is not a far ray, it describes its function value. If it is a directional ray, it describes the slope on that ray.

User Methods of RationalFunction

•  

  Defining morphisms and functions

  These methods are used to define morphisms or rational functions on a Cycle.

  •  

    restrict (C) → RationalFunction<Addition>

    
    

    Computes the restriction of this RationalFunction on a given Cycle. The cycle need not be contained in the DOMAIN of the function, the restriction will be computed on the intersection of the cycle and the DOMAIN.

    Parameters

    Cycle<Addition>

    C

    The new domain.

    Returns

    RationalFunction<Addition>

•  

  Visualization

  These methods are for visualization.

  •  

    VISUAL ()

    
    

    Visualizes the domain of the function. Works exactly as VISUAL of WeightedComplex, but has additional option @options

    Options

    String

    FunctionLabels

    If set to "show", textual function representations are diplayed on cones. False by default

•  

  insert_leaves (curve, nodes)

  
  

  Takes a RationalCurve and a list of node indices. Then inserts additional leaves (starting from N_LEAVES+1) at these nodes and returns the resulting RationalCurve object

  Parameters

  RationalCurve	curve	A RationalCurve object
  Vector<Int>	nodes	A list of node indices of the curve

•  

  matroid_coordinates_from_curve <Addition> (r) → Vector<Rational>

  
  

  Takes a rational curve and converts it into the corresponding matroid coordinates in the moduli space of rational curves (including the leading 0 for a ray!)

  Type Parameters

  Addition

  Min or Max, i.e. which coordinates to use.

  Parameters

  RationalCurve

  r

  A rational n-marked curve

  Returns

  Vector<Rational>

  The matroid coordinates, tropically homogeneous and with leading coordinate

•  

  rational_curve_from_cone (X, n_leaves, coneIndex) → RationalCurve

  
  

  This takes a weighted complex X that is supposed to be of the form M_0,n x Y for some Y (It assumes that M_0,n occupies the first coordinates) and an index of a maximal cone of that complex. It then computes a rational curve corresponding to an interior point of that cone (ignoring the second component Y)

  Parameters

  Cycle<Addition>	X	A weighted complex of the form M_0,n x Y
  Int	n_leaves	The n in M_0,n. Needed to determine the dimension of the M_0,n component
  Int	coneIndex	The index of the maximal cone

  Returns

  RationalCurve

  c The curve corresponding to an interior point

•  

  rational_curve_from_matroid_coordinates <Addition> (v) → RationalCurve

  
  

  Takes a vector from Q^((n-1) over 2) that lies in M_0,n (in its matroid coordinates) and computes the corresponding rational curve. In the isomorphism of the metric curve space and the moduli coordinates the last leaf is considered as the special leaf

  Type Parameters

  Addition

  Min or Max (i.e. what are the matroid coordinates using)

  Parameters

  Vector<Rational>

  v

  A vector in the moduli space (WITH leading coordinate).

  Returns

  RationalCurve

•  

  rational_curve_from_metric (v) → RationalCurve

  
  

  Takes a vector from Q^(n over 2) that describes an n-marked rational abstract curve as a distance vector between its leaves. It then computes the curve corresponding to this vector.

  Parameters

  Vector<Rational>

  v

  A vector of length (n over 2). Its entries are interpreted as the distances d(i,j) ordered lexicographically according to i,j. However, they need not be positive, as long as v is equivalent to a proper metric modulo leaf lengths.

  Returns

  RationalCurve

•  

  rational_curve_from_rays <Addition> (rays) → RationalCurve

  
  

  This takes a matrix of rays of a given cone that is supposed to lie in a moduli space M_0,n and computes the rational curve corresponding to an interior point. More precisely, if there are k vertices in homogeneous coordinates, it computes 1/k * (sum of these vertices), then it adds each directional ray. It then returns the curve corresponding to this point

  Type Parameters

  Addition

  Min or Max, where the coordinates live.

  Parameters

  Matrix<Rational>

  rays

  The rays of the cone, in tropical homogeneous coordinates.

  Returns

  RationalCurve

  c The curve corresponding to an interior point

•  

  rational_curve_immersion <Addition> (delta, type) → Cycle<Addition>

  
  

  This function creates an embedding of a rational tropical curve using a given abstract curve and degree

  Type Parameters

  Addition

  Min or Max

  Parameters

  Matrix<Rational>	delta	The degree of the curve in tropical projectve coordinates without leading coordinate. The number of rows should correspond to the number of leaves of type and the number of columns is the dimension of the space in which the curve should be realized
  RationalCurve	type	An abstract rational curve

  Returns

  Cycle<Addition>

  The corresponding immersed complex. The position of the curve is determined by the first node, which is always placed at the origin

•  

  rational_curve_list_from_matroid_coordinates <Addition> (m) → RationalCurve

  
  

  Takes a matrix whose rows are elements in the moduli space M_0,n in matroid coordinates. Returns a list, where the i-th element is the curve corr. to the i-th row in the matrix

  Type Parameters

  Addition

  Mir or Max (i.e. what are the matroid coordinates using

  Parameters

  Matrix<Rational>

  m

  A list of vectors in the moduli space (with leading coordinate).

  Returns

  RationalCurve

  : An array of RationalCurves

•  

  rational_curve_list_from_metric (m) → RationalCurve

  
  

  Takes a matrix whose rows are metrics of rational n-marked curves. Returns a list, where the i-th element is the curve corr. to the i-th row in the matrix

  Parameters

  Matrix<Rational>

  m

  Returns

  RationalCurve

  : An array of RationalCurves

•  

  sum_curves (An, v) → RationalCurve

  
  

  This function takes a vector of coefficients a_i and a list of RationalCurves c_i and computes sum(a_i * c_i). In particular, it also checks, whether the result lies in M_0,n. If not, it returns undef

  Parameters

  RationalCurve	An	arbitrary list of RationalCurve objects
  Vector<Rational>	v	A list of coefficients. Superfluous coefficients are ignored, missing ones replaced by +1(!)

  Returns

  RationalCurve

  The linear combination of the curves defined by the coefficients or undef, if the result is not in M_0,n. The history of the operation is kept in INPUT_SETS and INPUT_COEFFS

•  

  testFourPointCondition (v) → Int

  
  

  Takes a metric vector in Q^{(n over 2)} and checks whether it fulfills the four-point condition, i.e. whether it lies in M_0,n. More precisely it only needs to be equivalent to such a vector

  Parameters

  Vector<Rational>

  v

  The vector to be checked

  Returns

  Int	A quadruple (array) of indices, where the four-point condition is violated or an empty list, if the vector is indeed in M_0,n

•  

  affine_transform (C, M, T) → Cycle<Addition>

  
  

  Computes the affine transform of a cycle under an affine linear map. This function assumes that the map is a lattice isomorphism on the cycle, i.e. no push-forward computations are performed, in particular the weights remain unchanged

  Parameters

  Cycle<Addition>	C	a tropical cycle
  Matrix<Rational>	M	The transformation matrix. Should be given in tropical projective coordinates and be homogeneous, i.e. the sum over all rows should be the same.
  Vector<Rational>	T	The translate. Optional and zero vector by default. Should be given in tropical projective coordinates (but without leading coordinate for vertices or rays). If you only want to shift a cycle, use shift_cycle.

  Returns

  Cycle<Addition>

  The transform M*C + T

•  

  affine_transform (C, M) → Cycle<Addition>

  
  

  Computes the affine transform of a cycle under an affine linear map. This function assumes that the map is a lattice isomorphism on the cycle, i.e. no push-forward computations are performed, in particular the weights remain unchanged

  Parameters

  Cycle<Addition>	C	a tropical cycle
  Morphism<Addition>	M	A morphism. Should be defined via MATRIX and TRANSLATE, though its DOMAIN will be ignored.

  Returns

  Cycle<Addition>

  The transform M(C)

•  

  cartesian_product (cycles) → Cycle

  
  

  Computes the cartesian product of a set of cycles. If any of them has weights, so will the product (all non-weighted cycles will be treated as if they had constant weight 1)

  Parameters

  Cycle

  cycles

  a list of Cycles

  Returns

  Cycle

  The cartesian product. Note that the representation is noncanonical, as it identifies the product of two projective tori of dimensions d and e with a projective torus of dimension d+e by dehomogenizing and then later rehomogenizing after the first coordinate.

•  

  check_cycle_equality (X, Y, check_weights) → Bool

  
  

  This takes two pure-dimensional polyhedral complexes and checks if they are equal i.e. if they have the same lineality space, the same rays (modulo lineality space) and the same cones. Optionally, it can also check if the weights are equal

  Parameters

  Cycle<Addition>	X	A weighted complex
  Cycle<Addition>	Y	A weighted complex
  Bool	check_weights	Whether the algorithm should check for equality of weights. This parameter is optional and true by default

  Returns

  Bool	Whether the cycles are equal

•  

  coarsen (complex, testFan) → Cycle<Addition>

  
  

  Takes a tropical variety on which a coarsest polyhedral structure exists and computes this structure.

  Parameters

  Cycle<Addition>	complex	A tropical variety which has a unique coarsest polyhedral structre
  Bool	testFan	(Optional, FALSE by default). Whether the algorithm should perform some consistency checks on the result. If true, it will check the following: - That equivalence classes of cones have convex support - That all equivalence classes have the same lineality space If any condition is violated, the algorithm throws an exception Note that it does not check whether equivalence classes form a fan This can be done via fan::check_fan afterwards, but it is potentially slow.

  Returns

  Cycle<Addition>

  The corresponding coarse complex. Throws an exception if testFan = True and consistency checks fail.

•  

  contains_point (A, point) → Bool

  
  

  Takes a weighted complex and a point and computed whether that point lies in the complex

  Parameters

  Cycle	A	weighted complex
  Vector<Rational>	point	An arbitrary vector in the same ambient dimension as complex. Given in tropical projective coordinates with leading coordinate.

  Returns

  Bool	Whether the point lies in the support of complex

•  

  fan_decomposition (C) → Cycle<Addition>

  
  

  This computes the local fans at all (nonfar) vertices of a tropical cycle

  Parameters

  Cycle<Addition>

  C

  A tropical cycle

  Returns

  Cycle<Addition>

  A list of local cycles

•  

  insert_rays (F, R) → Cycle<Addition>

  
  

  Takes a cycle and a list of rays/vertices in tropical projective coordinates with leading coordinate and triangulates the fan such that it contains these rays

  Parameters

  Cycle<Addition>	F	A cycle (not necessarily weighted).
  Matrix<Rational>	R	A list of normalized vertices or rays Note that the function will NOT subdivide the lineality space, i.e. rays that are equal to an existing ray modulo lineality space will be ignored.

  Returns

  Cycle<Addition>

  A triangulation of F that contains all the original rays of F plus the ones in R

•  

  intersect_container (cycle, container, forceLatticeComputation) → Cycle

  
  

  Takes two Cycles and computes the intersection of both. The function relies on the fact that the second cycle contains the first cycle to compute the refinement correctly The function copies WEIGHTS, LATTICE_BASES and LATTICE_GENERATORS in the obvious manner if they exist.

  Parameters

  Cycle	cycle	An arbitrary Cycle
  Cycle	container	A cycle containing the first one (as a set) Doesn't need to have any weights and its tropical addition is irrelevant.
  Bool	forceLatticeComputation	Whether the properties LATTICE_BASES and LATTICE_GENERATORS of cycle should be computed before refining. False by default.

  Returns

  Cycle

  The intersection of both complexes (whose support is equal to the support of cycle). It uses the same tropical addition as cycle.

•  

  recession_fan (complex) → Cycle

  
  

  Computes the recession fan of a tropical variety. WARNING: This is a highly experimental function. If it works at all, it is likely to take a very long time for larger objects.

  Parameters

  Cycle

  complex

  A tropical variety

  Returns

  Cycle

  A tropical fan, the recession fan of the complex

•  

  set_theoretic_intersection (A, B) → fan::PolyhedralComplex

  
  

  Computes the set-theoretic intersection of two cycles and returns it as a polyhedral complex. The cycles need not use the same tropical addition

  Parameters

  Cycle	A
  Cycle	B

  Returns

  fan::PolyhedralComplex

  The set-theoretic intersection of the supports of A and B

•  

  shift_cycle (C, T) → Cycle<Addition>

  
  

  Computes the shift of a tropical cycle by a given vector

  Parameters

  Cycle<Addition>	C	a tropical cycle
  Vector<Rational>	T	The translate. Optional and zero vector by default. Should be given in tropical projective coordinates (but without leading coordinate for vertices or rays).

  Returns

  Cycle<Addition>

  The shifted cycle

•  

  skeleton_complex (C, k, preserveRays) → Cycle<Addition>

  
  

  Takes a polyhedral complex and computes the k-skeleton. Will return an empty cycle, if k is larger then the dimension of the given complex or smaller than 0.

  Parameters

  Cycle<Addition>	C	A polyhedral complex.
  Int	k	The dimension of the skeleton that should be computed
  Bool	preserveRays	When true, the function assumes that all rays of the fan remain in the k-skeleton, so it just copies the VERTICES, instead of computing an irredundant list. By default, this property is false.

  Returns

  Cycle<Addition>

  The k-skeleton (without any weights, except if k is the dimension of C

•  

  triangulate_cycle (F) → Cycle<Addition>

  
  

  Takes a cycle and computes a triangulation

  Parameters

  Cycle<Addition>

  F

  A cycle (not necessarily weighted)

  Returns

  Cycle<Addition>

  A simplicial refinement of F

•  

  projection_map <Addition> (n, s) → Morphism<Addition>

  
  

  This creates a linear projection from the projective torus of dimension n to a given set of coordinates.

  Type Parameters

  Addition

  Min or Max

  Parameters

  Int	n	The dimension of the projective torus which is the domain of the projection.
  Set<Int>	s	The set of coordinaes to which the map should project. Should be a subset of (0,..,n)

  Returns

  Morphism<Addition>

  The projection map.

•  

  projection_map (n, m) → Morphism

  
  

  This computes the projection from a projective torus of given dimension to a projective torus of lower dimension which lives on the first coordinates

  Parameters

  Int	n	The dimension of the larger torus
  Int	m	The dimension of the smaller torus

  Returns

  Morphism

  The projection map

•  

  affine_linear_space <Addition> (lineality, translate, weight) → Cycle<Addition>

  
  

  This creates a true affine linear space.

  Type Parameters

  Addition

  Min or Max

  Parameters

  Matrix<Rational>	lineality	(Row) generators of the lineality space, in tropical homogeneous coordinates, but without the leading zero
  Vector<Rational>	translate	Optional. The vertex of the space. By default this is the origin
  Integer	weight	Optional. The weight of the space. By default, this is 1.

  Returns

  Cycle<Addition>

•  

  cross_variety <Addition> (n, k, h, weight) → Cycle<Addition>

  
  

  This creates the k-skeleton of the tropical variety dual to the cross polytope

  Type Parameters

  Addition

  Min or Max

  Parameters

  Int	n	The (projective) ambient dimension
  Int	k	The (projective) dimension of the variety.
  Rational	h	Optional, 1 by default. It is a nonnegative number, describing the height of the one interior lattice point of the cross polytope.
  Integer	weight	Optional, 1 by default. The (global) weight of the variety

  Returns

  Cycle<Addition>

  The k-skeleton of the tropical hypersurface dual to the cross polytope. It is a smooth (for weight 1), irreducible (for h > 0) variety, which is invariant under reflection.

•  

  empty_cycle <Addition> (ambient_dim) → Cycle

  
  

  Creates the empty cycle in a given ambient dimension (i.e. it will set the property PROJECTIVE_AMBIENT_DIM.

  Type Parameters

  Addition

  Max or Min

  Parameters

  Int

  ambient_dim

  The ambient dimension

  Returns

  Cycle

  The empty cycle

•  

  halfspace_subdivision <Addition> (a, g, The) → Cycle

  
  

  Creates a subdivision of the tropical projective torus along an affine hyperplane into two halfspaces. This hyperplane is defined by an equation gx = a

  Type Parameters

  Addition

  Max or Min

  Parameters

  Rational	a	The constant coefficient of the equation
  Vector<Rational>	g	The linear coefficients of the equation Note that the equation must be homogeneous in the sense that (1,..1) is in its kernel, i.e. all entries of g add up to 0.
  Integer	The	(constant) weight this cycle should have

  Returns

  Cycle

  The halfspace subdivision

•  

  matroid_ring_cycle <Addition> (M, scale) → MatroidRingCycle<Addition>

  
  

  This creates the cycle corresponding to a given matroid. It automatically computes its representation in the basis of nested matroids. This is equivalent to using the constructor of MatroidRingCycle.

  Type Parameters

  Addition

  The tropical addition. Either Min or Max.

  Parameters

  matroid::Matroid	M	A matroid
  Int	scale	An optional linear coefficient. The resulting cycle will be scale*B(M) in the ring of matroids.

  Returns

  MatroidRingCycle<Addition>

  
  

  Example:
•  

  orthant_subdivision <Addition> (point, chart, weight)

  
  

  Creates the orthant subdivision around a given point on a given chart, i.e. the corresponding affine chart of this cycle consists of all 2^n fulldimensional orthants

  Type Parameters

  Addition

  Min or Max

  Parameters

  Vector<Rational>	point	The vertex of the subdivision. Should be given in tropical homogeneous coordinates with leading coordinate.
  Int	chart	On which chart the cones should be orthants, 0 by default.
  Integer	weight	The constant weight of the cycle, 1 by default.

•  

  point_collection <Addition> (points, weights) → Cycle

  
  

  Creates a cycle consisting of a collection of points with given weights

  Type Parameters

  Addition

  Max or Min

  Parameters

  Matrix<Rational>	points	The points, in tropical homogeneous coordinates (though not with leading ones for vertices).
  Vector<Integer>	weights	The list of weights for the points

  Returns

  Cycle

  The point collection.

•  

  projective_torus <Addition> (n, w) → Cycle

  
  

  Creates the tropical projective torus of a given dimension. In less fancy words, the cycle is the complete complex of given (tropical projective) dimension n, i.e. Rⁿ

  Type Parameters

  Addition

  Max or Min.

  Parameters

  Int	n	The tropical projective dimension.
  Integer	w	The weight of the cycle. Optional and 1 by default.

  Returns

  Cycle

  The tropical projective torus.

•  

  uniform_linear_space <Addition> (n, k, weight) → Cycle

  
  

  Creates the linear space of the uniform matroid of rank k+1 on n+1 variables.

  Type Parameters

  Addition

  A The tropical addition (min or max)

  Parameters

  Int	n	The ambient (projective) dimension.
  Int	k	The (projective dimension of the fan.
  Integer	weight	The global weight of the cycle. 1 by default.

  Returns

  Cycle

  A tropical linear space.

•  

  is_empty ()

  
  

  This tests wheter a cycle is the empty cycle.

•  

  divisor (C, F) → Cycle

  
  

  This function computes the divisor of one or more rational functions on a tropical cycle.

  Parameters

  Cycle	C	A tropical cycle
  RationalFunction	F	An arbitrary list of rational functions (r_1,...r_n). The DOMAIN of r_i should contain the support of r_{i-1} * ... * r_1 * C. Note that using the ^-operator on these rational functions is allowed and will result in applying the corresponding function several times.

  Returns

  Cycle

  The divisor r_n * ... * r_1 * C

•  

  divisor_nr (C, F) → Cycle

  
  

  This function computes the divisor of one or more rational functions on a tropical cycle. It should only be called, if the DOMAIN of all occuring cycles is the cycle itself. This function will be faster than divisor, since it computes no refinements.

  Parameters

  Cycle	C	A tropical cycle
  RationalFunction	F	An arbitrary list of rational functions (r_1,...r_n). The DOMAIN of each function should be equal (in terms of VERTICES and MAXIMAL_POLYTOPES) to the cycle. Note that using the ^-operator on these rational functions is allowed and will result in applying the corresponding function several times.

  Returns

  Cycle

  The divisor r_n * ... * r_1 * C

•  

  piecewise_divisor (F, cones, coefficients) → Cycle<Addition>

  
  

  Computes a divisor of a linear sum of certain piecewise polynomials on a simplicial fan.

  Parameters

  Cycle<Addition>	F	A simplicial fan without lineality space in non-homog. coordinates
  IncidenceMatrix	cones	A list of cones of F (not maximal, but all of the same dimension). Each cone t corresponds to a piecewise polynomial psi_t, defined by subsequently applying the rational functions that are 1 one exactly one ray of t and 0 elsewhere. Note that cones should refer to indices in SEPARATED_VERTICES, which may have a different order
  Vector<Integer>	coefficients	A list of coefficients a_t corresponding to the cones.

  Returns

  Cycle<Addition>

  The divisor sum_t a_t psi_t * F

•  

  hurwitz_cycle <Addition> (k, degree, points) → Cycle<Addition>

  
  

  This function computes the Hurwitz cycle H_k(x), x = (x_1,...,x_n)

  Type Parameters

  Addition

  Min or Max, where the coordinates live.

  Parameters

  Int	k	The dimension of the Hurwitz cycle, i.e. the number of moving vertices
  Vector<Int>	degree	The degree x. Should add up to 0
  Vector<Rational>	points	Optional. Should have length n-3-k. Gives the images of the fixed vertices (besides 0). If not given all fixed vertices are mapped to 0 and the function computes the recession fan of H_k(x)

  Options

  Bool

  Verbose

  If true, the function outputs some progress information. True by default.

  Returns

  Cycle<Addition>

  H_k(x), in homogeneous coordinates

•  

  hurwitz_marked_cycle <Addition> (k, degree, pullback_points) → Cycle<Addition>

  
  

  Computes the marked k-dimensional tropical Hurwitz cycle H_k(degree)

  Type Parameters

  Addition

  Min or Max

  Parameters

  Int	k	The dimension of the Hurwitz cycle
  Vector<Int>	degree	The degree of the covering. The sum over all entries should be 0 and if n := degree.dim, then 0 <= k <= n-3
  Vector<Rational>	pullback_points	The points p_i that should be pulled back to determine the Hurwitz cycle (in addition to 0). Should have length n-3-k. If it is not given, all p_i are by default equal to 0 (same for missing points)

  Returns

  Cycle<Addition>

  The marked Hurwitz cycle H~_k(degree)

•  

  hurwitz_pair <Addition> (k, degree, points) → List

  
  

  This function computes hurwitz_subdivision and hurwitz_cycle at the same time, returning the result in an array

  Type Parameters

  Addition

  Min or Max, where the coordinates live.

  Parameters

  Int	k	The dimension of the Hurwitz cycle, i.e. the number of moving vertices
  Vector<Int>	degree	The degree x. Should add up to 0
  Vector<Rational>	points	Optional. Should have length n-3-k. Gives the images of the fixed vertices (besides 0). If not given all fixed vertices are mapped to 0 and the function computes the subdivision of M_0,n containing the recession fan of H_k(x)

  Options

  Bool

  Verbose

  If true, the function outputs some progress information. True by default.

  Returns

  List	( Cycle subdivision of M_0,n, Cycle Hurwitz cycle )

•  

  hurwitz_pair_local <Addition> (k, degree, local_curve)

  
  

  Does the same as hurwitz_pair, except that no points are given and the user can give a RationalCurve object representing a ray. If given, the computation will be performed locally around the ray.

  Type Parameters

  Addition

  Min or Max, where the coordinates live.

  Parameters

  Int	k
  Vector<Int>	degree
  RationalCurve	local_curve

  Options

  Bool

  Verbose

  If true, the function outputs some progress information. True by default.

•  

  hurwitz_subdivision <Addition> (k, degree, points) → Cycle

  
  

  This function computes a subdivision of M_0,n containing the Hurwitz cycle H_k(x), x = (x_1,...,x_n) as a subfan. If k = n-4, this subdivision is the unique coarsest subdivision fulfilling this property

  Type Parameters

  Addition

  Min or Max, where the coordinates live.

  Parameters

  Int	k	The dimension of the Hurwitz cycle, i.e. the number of moving vertices
  Vector<Int>	degree	The degree x. Should add up to 0
  Vector<Rational>	points	Optional. Should have length n-3-k. Gives the images of the fixed vertices (besides the first one, which always goes to 0) as elements of R. If not given, all fixed vertices are mapped to 0 and the function computes the subdivision of M_0,n containing the recession fan of H_k(x)

  Options

  Bool

  Verbose

  If true, the function outputs some progress information. True by default.

  Returns

  Cycle

  A subdivision of M_0,n

•  

  degree (A) → Integer

  
  

  Computes the degree of a tropical variety as the total weight of the 0-dimensional intersection product obtained by intersecting with the complementary uniform linear space.

  Parameters

  Cycle

  A

  tropical cycle

  Returns

  Integer

  The degree

•  

  intersect (X, Y) → Cycle

  
  

  Computes the intersection product of two tropical cycles in the projective torus Use intersect_check_transversality to check for transversal intersections

  Parameters

  Cycle	X	A tropical cycle
  Cycle	Y	A tropical cycle, living in the same ambient space as X

  Returns

  Cycle

  The intersection product

•  

  intersect_check_transversality (X, Y, ensure_transversality) → List

  
  

  Computes the intersection product of two tropical cycles in R^n and tests whether the intersection is transversal (in the sense that the cycles intersect set-theoretically in the right dimension).

  Parameters

  Cycle	X	A tropical cycle
  Cycle	Y	A tropical cycle, living in the same space as X
  Bool	ensure_transversality	Whether non-transversal intersections should not be computed. Optional and false by default. If true, returns the zero cycle if it detects a non-transversal intersection

  Returns

  List	( Cycle intersection product, Bool is_transversal). Intersection product is a zero cycle if ensure_transversality is true and the intersection is not transversal. is_transversal is false if the codimensions of the varieties add up to more than the ambient dimension.

•  

  intersect_in_smooth_surface (surface, A, B) → Cycle<Addition>

  
  

  Computes the intersection product of two cycles in a smooth surface

  Parameters

  Cycle<Addition>	surface	A smooth surface
  Cycle<Addition>	A	any cycle in the surface
  Cycle<Addition>	B	any cycle in the surface

  Returns

  Cycle<Addition>

  The intersection product of A and B in the surface

•  

  point_functions <Addition> (A) → RationalFunction

  
  

  Constructs a list of rational functions that cut out a single point in the projective torus

  Type Parameters

  Addition

  Min or Max. Determines the type of the rational functions.

  Parameters

  Vector<Rational>

  A

  point in the projective torus, given in tropical homogeneous coordinates, but without leading coordinate.

  Returns

  RationalFunction

  . A perl array of rational functions of the form (v_i*x_0 + x_i)/(x_0), i = 1,..,n

•  

  pullback (m, r) → RationalFunction

  
  

  This computes the pullback of a rational function via a morphism Due to the implementation of composition of maps, the DOMAIN of the rational function need not be contained in the image of the morphism The pullback will be defined in the preimage of the domain.

  Parameters

  Morphism	m	A morphism.
  RationalFunction	r	A rational function.

  Returns

  RationalFunction

  The pullback m*r.

•  

  cutting_functions (F, weight_aim) → Matrix<Rational>

  
  

  Takes a weighted complex and a list of desired weights on its codimension one faces and computes all possible rational functions on (this subdivision of ) the complex

  Parameters

  Cycle<Addition>	F	A tropical variety, assumed to be simplicial.
  Vector<Integer>	weight_aim	A list of weights, whose length should be equal to the number of CODIMENSION_ONE_POLYTOPES. Gives the desired weight on each codimension one face

  Returns

  Matrix<Rational>

  The space of rational functions defined on this particular subdivision. Each row is a generator. The columns correspond to values on SEPARATED_VERTICES and LINEALITY_SPACE, except the last one, which is either 0 (then this function cuts out zero and can be added to any solution) or non-zero (then normalizing this entry to -1 gives a function cutting out the desired weights on the codimension one skeleton Note that the function does not test if these generators actually define piecewise linear functions, as it assumes the cycle is simplicial

•  

  simplicial_diagonal_system (fan) → Matrix<Rational>

  
  

  This function computes the inhomogeneous version of simplicial_piecewise_system in the sense that it computes the result of the above mentioned function (i.e. which coefficients for the piecewise polynomials yield the zero divisor) and adds another column at the end where only the entries corresponding to the diagonal cones are 1, the rest is zero. This can be seen as asking for a solution to the system that cuts out the diagonal (all solutions whose last entry is 1)

  Parameters

  Cycle<Addition>

  fan

  . A simplicial fan without lineality space.

  Returns

  Matrix<Rational>

•  

  simplicial_piecewise_system (F) → Matrix<Rational>

  
  

  This function takes a d-dimensional simplicial fan F and computes the linear system defined in the following way: For each d-dimensional cone t in the diagonal subdivision of FxF, let psi_t be the piecewise polynomial defined by subsequently applying the rational functions that are 1 one exactly one ray of t and 0 elsewhere. Now for which coefficients a_t is sum_t a_t psi_t * (FxF) = 0?

  Parameters

  Cycle<Addition>

  F

  A simplicial fan without lineality space

  Returns

  Matrix<Rational>

  The above mentioned linear system. The rows are equations, the columns correspond to d-dimensional cones of FxF in the order given by skeleton_complex(simplicial_with_diagonal(F), d, 1)

•  

  simplicial_with_diagonal (F) → Cycle<Addition>

  
  

  This function takes a simplicial fan F (without lineality space) and computes the coarsest subdivision of F x F containing all diagonal rays (r,r)

  Parameters

  Cycle<Addition>

  F

  A simplicial fan without lineality space.

  Returns

  Cycle<Addition>

  The product complex FxF subdivided such that it contains all diagonal rays

•  

  lattice_index (m) → Integer

  
  

  This computes the index of a lattice in its saturation.

  Parameters

  Matrix<Integer>

  m

  A list of (row) generators of the lattice.

  Returns

  Integer

  The index of the lattice in its saturation.

•  

  randomInteger (max_arg, n) → Array<Integer>

  
  

  Returns n random integers in the range 0.. (max_arg-1),inclusive Note that this algorithm is not optimal for real randomness: If you change the range parameter and then change it back, you will usually get the exact same sequence as the first time

  Parameters

  Int	max_arg	The upper bound for the random integers
  Int	n	The number of integers to be created

  Returns

  Array<Integer>

•  

  lines_in_cubic (p) → LinesInCubic<Addition>

  
  

  This takes either: - A homogeneous polynomial of degree 3 in 4 variables or - A polynomial of degree 3 in 3 variables and computes the corresponding cubic and finds all tropical lines and families thereof in the cubic. The result is returned as a LinesInCubic object. Note that the function has some heuristics for recognizing families, but might still return a single family as split up into two.

  Parameters

  Polynomial<TropicalNumber<Addition>>

  p

  A homogeneous tropical polynomial of degree 3 in four variables.

  Returns

  LinesInCubic<Addition>

•  

  local_codim_one (complex, face) → Cycle<Addition>

  
  

  This takes a weighted complex and an index of one of its codimension one faces (The index is in CODIMENSION_ONE_POLYTOPES) and computes the complex locally restricted to that face

  Parameters

  Cycle<Addition>	complex	An arbitrary weighted complex
  Int	face	An index of a face in CODIMENSION_ONE_POLYTOPES

  Returns

  Cycle<Addition>

  The complex locally restricted to the given face

•  

  local_point (complex, v) → Cycle<Addition>

  
  

  This takes a weighted complex and an arbitrary vertex in homogeneous coordinates (including the leading coordinate) that is supposed to lie in the support of the complex. It then refines the complex such that the vertex is a cell in the polyhedral structure and returns the complex localized at this vertex

  Parameters

  Cycle<Addition>	complex	An arbitrary weighted complex
  Vector<Rational>	v	A vertex in homogeneous coordinates and with leading coordinate. It should lie in the support of the complex (otherwise an error is thrown)

  Returns

  Cycle<Addition>

  The complex localized at the vertex

•  

  local_restrict (complex, cones) → Cycle<Addition>

  
  

  This takes a tropical variety and an IncidenceMatrix describing a set of cones (not necessarily maximal ones) of this variety. It will then create a variety that contains all compatible maximal cones and is locally restricted to the given cone set.

  Parameters

  Cycle<Addition>	complex	An arbitrary weighted complex
  IncidenceMatrix	cones	A set of cones, indices refer to VERTICES

  Returns

  Cycle<Addition>

  The same complex, locally restricted to the given cones

•  

  local_vertex (complex, ray) → Cycle<Addition>

  
  

  This takes a weighted complex and an index of one of its vertices (the index is to be understood in VERTICES) It then localizes the variety at this vertex. The index should never correspond to a far vertex in a complex, since this would not be a cone

  Parameters

  Cycle<Addition>	complex	An arbitrary weighted complex
  Int	ray	The index of a ray/vertex in RAYS

  Returns

  Cycle<Addition>

  The complex locally restricted to the given vertex

•  

  star_at_point (C, v) → Cycle<Addition>

  
  

  Computes the Star of a tropical cycle at an arbitrary point in its support

  Parameters

  Cycle<Addition>	C	a tropical cycle
  Vector<Rational>	v	A point, given in tropical projective coordinates with leading coordinate and which should lie in the support of C

  Returns

  Cycle<Addition>

  The Star of C at v (Note that the subdivision may be finer than a potential coarsest structure

•  

  star_at_vertex (C, i) → Cycle<Addition>

  
  

  Computes the Star of a tropical cycle at one of its vertices.

  Parameters

  Cycle<Addition>	C	a tropical cycle
  Int	i	The index of a vertex in VERTICES, which should not be a ray

  Returns

  Cycle<Addition>

  The Star of C at the vertex

•  

  matroid_ring_linear_space (L) → Matrix<Rational>

  
  

  Given a list of MatroidRingCycle objects (of the same rank r, on the same ground set), computes a matrix that represents the linear space spanned by these cycles in the rank r graded part of the matroid ring. Rows correspond to the cycles, columns correspond to the set of all the nested matroid occuring in all basis presentations of the cycles. Entries are linear coefficients.

  Parameters

  MatroidRingCycle

  L

  A list of matroid ring cycles.

  Returns

  Matrix<Rational>

  A matrix representation of the linear space spanned by L

  
  

  Example:
•  

  matroid_ring_product (A, B) → MatroidRingCycle

  
  

  Computes the product of two matroid ring cycles.

  Parameters

  MatroidRingCycle	A
  MatroidRingCycle	B

  Returns

  MatroidRingCycle

  A * B

•  

  matroid_ring_sum (A, B) → MatroidRingCycle

  
  

  Computes the sum of two matroid ring cycles

  Parameters

  MatroidRingCycle	A
  MatroidRingCycle	B

  Returns

  MatroidRingCycle

  A + B

•  

  zero_in_matroid_ring <Addition> (n) → MatroidRingCycle

  
  

  Constructs the zero element of the matroid ring.

  Type Parameters

  Addition

  The tropical Addition, either Min or Max.

  Parameters

  Int

  n

  The size of the ground set.

  Returns

  MatroidRingCycle

  
  

  Example:

•  

  is_smooth (a) → List

  
  

  Takes a weighted fan and returns if it is smooth (i.e. isomorphic to a Bergman fan B(M)/L for some matroid M) or not. The algorithm works for fans of dimension 0,1,2 and codimension 0,1! For other dimensions the algorithm could give an answer but it is not guaranteed.

  Parameters

  Cycle<Addition>

  a

  tropical fan F

  Returns

  List

  ( Int s, Matroid M, Morphism<Addition> A ). If s=1 then F is smooth, the corresponding matroid fan is Z-isomorphic to the matroid fan associated to M. The Z-isomorphism is given by A, i.e. B(M)/L = affine_transform(F,A) If s=0, F is not smooth. If s=2 the algorithm is not able to determine if F is smooth or not.

•  

  matroid_fan <Addition> (m) → Cycle

  
  

  Uses an algorithm by Felipe Rincón to compute the matroidal fan of a given matroid. If you have a matrix at hand that represents this matroid, it is recommended to call this function with that matrix as an argument - it is significantly faster.

  Type Parameters

  Addition

  Min or Max - determines the coordinates.

  Parameters

  matroid::Matroid

  m

  A matroid

  Returns

  Cycle

  The matroidal fan or Bergman fan of the matroid.

•  

  matroid_fan <Addition> (m) → Cycle

  
  

  Uses an algorithm by Felipe Rincón to compute the bergman fan of the column matroid of the given matrix. Calling the function in this manner is significantly faster than calling it on the matroid.

  Type Parameters

  Addition

  Min or Max - determines the coordinates.

  Parameters

  Matrix<Rational>

  m

  A matrix, whose column matroid is considered.

  Returns

  Cycle

  The matroidal fan or Bergman fan of the matroid.

•  

  matroid_fan_from_flats <Addition> (A) → Cycle<Addition>

  
  

  Computes the fan of a matroid in its chains-of-flats subdivision. Note that this is potentially very slow for large matroids.

  Type Parameters

  Addition

  Min or max, determines the matroid fan coordinates.

  Parameters

  matroid::Matroid

  A

  matroid. Should be loopfree.

  Returns

  Cycle<Addition>

•  

  matroid_from_fan (A) → matroid::Matroid

  
  

  Takes the bergman fan of a matroid and reconstructs the corresponding matroid The fan has to be given in its actual matroid coordinates, not as an isomorphic transform. The actual subdivision is not relevant.

  Parameters

  Cycle<Addition>

  A

  tropical cycle, the Bergman fan of a matroid

  Returns

  matroid::Matroid

•  

  count_mn_cones (n, k) → Integer

  
  

  Computes the number of k-dimensional cones of the tropical moduli space M_0,n

  Parameters

  Int	n	The number of leaves. Should be >= 3
  Int	k	The number of bounded edges. This argument is optional and n-3 by default

  Returns

  Integer

  The number of k-dimensional cones of M_0,n

•  

  count_mn_rays (n) → Integer

  
  

  Computes the number of rays of the tropical moduli space M_0,n

  Parameters

  Int

  n

  The number of leaves. Should be >= 3

  Returns

  Integer

  The number of rays

•  

  evaluation_map <Addition> (n, Delta, i) → Morphism<Addition>

  
  

  This creates the i-th evaluation function on M_0,n^(lab)(R^r,Delta) (which is actually realized as M_0,(n+|Delta|) x R^r and can be created via space_of_stable_maps).

  Type Parameters

  Addition

  Min or Max

  Parameters

  Int	n	The number of marked (contracted) points
  Matrix<Rational>	Delta	The directions of the unbounded edges (given as row vectors in tropical projective coordinates without leading coordinate, i.e. have r+1 columns)
  Int	i	The index of the marked point that should be evaluated. Should lie in between 1 and n Note that the i-th marked point is realized as the |Delta|+i-th leaf in M_0,(n+|Delta|) and that the R^r - coordinate is interpreted as the position of the n-th leaf. In particular, ev_n is just the projection to the R^r-coordinates

  Returns

  Morphism<Addition>

  ev_i. Its domain is the ambient space of the moduli space as created by space_of_stable_maps. The target space is the tropical projective torus of dimension r

•  

  evaluation_map <Addition> (n, r, d, i) → Morphism<Addition>

  
  

  This creates the i-th evaluation function on M_0,n^(lab)(R^r,d) (which is actually realized as M_0,(n+d(r+1)) x R^r) This is the same as calling the function evaluation_map(Int,Int,Matrix<Rational>,Int) with the standard d-fold degree as matrix (i.e. each (inverted) unit vector of R^(r+1) occuring d times).

  Type Parameters

  Addition

  Min or Max

  Parameters

  Int	n	The number of marked (contracted) points
  Int	r	The dimension of the target space
  Int	d	The degree of the embedding. The direction matrix will be the standard d-fold directions, i.e. each unit vector (inverted for Max), occuring d times.
  Int	i	The index of the marked point that should be evaluated. i should lie in between 1 and n

  Returns

  Morphism<Addition>

  ev_i. Its domain is the ambient space of the moduli space as created by space_of_stable_maps. The target space is the tropical projective torus of dimension r

•  

  forgetful_map <Addition> (n, S) → Morphism

  
  

  This computes the forgetful map from the moduli space M_0,n to M_0,(n-|S|)

  Type Parameters

  Addition

  Min or Max

  Parameters

  Int	n	The number of leaves in the moduli space M_0,n
  Set<Int>	S	The set of leaves to be forgotten. Should be a subset of (1,..,n)

  Returns

  Morphism

  The forgetful map. It will identify the remaining leaves i_1,..,i_(n-|S|) with the leaves of M_0,(n-|S|) in canonical order. The domain of the morphism is the ambient space of the morphism in matroid coordinates, as created by m0n.

•  

  local_m0n <Addition> (R ...) → Cycle<Addition>

  
  

  Computes the moduli space M_0,n locally around a given list of combinatorial types. More precisely: It computes the weighted complex consisting of all maximal cones containing any of the given combinatorial types and localizes at these types This should only be used for curves of small codimension. What the function actually does, is that it combinatorially computes the cartesian products of M_0,v's, where v runs over the possible valences of vertices in the curves For max(v) <= 8 this should terminate in a reasonable time (depending on the number of curves) The coordinates are the same that would be produced by the function m0n

  Type Parameters

  Addition

  Min or Max, determines the coordinates

  Parameters

  RationalCurve

  R ...

  A list of rational curves (preferrably in the same M_0,n)

  Returns

  Cycle<Addition>

  The local complex

•  

  m0n <Addition> (n) → Cycle

  
  

  Creates the moduli space of abstract rational n-marked curves. Its coordinates are given as the coordinates of the bergman fan of the matroid of the complete graph on n-1 nodes (but not computed as such) The isomorphism to the space of curve metrics is obtained by choosing the last leaf as special leaf

  Type Parameters

  Addition

  Min or Max

  Parameters

  Int

  n

  The number of leaves. Should be at least 3

  Returns

  Cycle

  The tropical moduli space M_0,n

•  

  psi_class <Addition> (n, i) → Cycle

  
  

  Computes the i-th psi class in the moduli space of n-marked rational tropical curves M_0,n

  Type Parameters

  Addition

  Min or Max

  Parameters

  Int	n	The number of leaves in M_0,n
  Int	i	The leaf for which we want to compute the psi class ( in 1,..,n )

  Returns

  Cycle

  The corresponding psi class

•  

  psi_product <Addition> (n, exponents) → Cycle

  
  

  Computes a product of psi classes psi_1^k_1 * ... * psi_n^k_n on the moduli space of rational n-marked tropical curves M_0,n

  Type Parameters

  Addition

  Min or Max

  Parameters

  Int	n	The number of leaves in M_0,n
  Vector<Int>	exponents	The exponents of the psi classes k_1,..,k_n. If the vector does not have length n or if some entries are negative, an error is thrown

  Returns

  Cycle

  The corresponding psi class divisor

•  

  space_of_stable_maps <Addition> (n, d, r) → Cycle

  
  

  Creates the moduli space of stable maps of rational n-marked curves into a projective torus. It is given as the cartesian product of M_{0,n+d} and R^r, where n is the number of contracted leaves, d the number of non-contracted leaves and r is the dimension of the target torus. The R^r - coordinate is interpreted as the image of the last (n-th) contracted leaf. Due to the implementation of cartesian_product, the projective coordinates are non-canonical: Both M_{0,n+d} and R^r are dehomogenized after the first coordinate, then the product is taken and homogenized after the first coordinate again. Note that functions in a-tint will usually treat this space in such a way that the first d leaves are the non-contracted ones and the remaining n leaves are the contracted ones.

  Type Parameters

  Addition

  Min or Max. Determines the coordinates.

  Parameters

  Int	n	The number of contracted leaves
  Int	d	The number of non-contracted leaves
  Int	r	The dimension of the target space for the stable maps.

  Returns

  Cycle

  The moduli space of rational stable maps.

•  

  add_morphisms (f, g) → Morphism

  
  

  Computes the sum of two morphisms. Both DOMAINs should have the same support and the target spaces should have the same ambient dimension The domain of the result will be the common refinement of the two domains.

  Parameters

  Morphism	f
  Morphism	g

  Returns

  Morphism

•  

  visualize_in_surface ()

  
  

  This visualizes a surface in R^3 and an arbitrary list of (possibly non-pure) Cycle objects. A common bounding box is computed for all objects and a random color is chosen for each object (except the surface)

•  

  decomposition_polytope (A) → polytope::Polytope

  
  

  Computes the possible positive decompositions into irreducible subvarieties of the same weight positivity signature (i.e. the weight on a cone has to have the same sign as in the cycle) To be precise, it computes the irreducible varieties as rays of the weight cone (where the corresponding orthant is taken such that the weight vector of X lies in that orthant). It then computes the polytope of all positive linear combinations of those irreducible varieties that produce the original weight vector.

  Parameters

  Cycle

  A

  weighted complex

  Returns

  polytope::Polytope

•  

  weight_cone (X, negative)

  
  

  Takes a polyhedral complex and computes a weight cone, i.e. intersects the WEIGHT_SPACE with a chosen orthant (by default the positive orthant)

  Parameters

  Cycle	X	A polyhedral complex
  Set<int>	negative	A subset of the coordinates {0,..,N-1}, where N is the number of maximal cells of X. Determines the orthant to intersect the weight space with: All coordinates in the set are negative, the others positive If the set is not given, it is empty by default (i.e. we take the positive orthant)