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: fan, group, ideal, matroid, polytope, topaz

Objects

  •  

    This is an augmented version of FaceLattice, which for each cell in a covector decomposition also saves the corresponding type.

    derived from: graph::FaceLattice

    Properties of CovectorLattice

    •  

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

      •  
        COVECTORS: common::NodeMap<Directed, IncidenceMatrix<NonSymmetric>>

        Each node in the face lattice is a cell of a covector decomposition (of either the tropical torus or the tropical span of some points). This property maps each cell to the corresponding covector. A covector is encoded as an IncidenceMatrix, where rows correspond to coordinates and columns to POINTS.

    User Methods of CovectorLattice

    •  

      These methods are for visualization.

      •  
        VISUAL ()

        Visualizes the covector lattice. This works the same as a visualization of a Hasse diagram except that by default, covectors are displayed. This can be turned off by the option Covectors=>"hidden"

  •  

    A tropical cycle is a weighted, balanced, pure polyhedral complex. It is given as a polyhedral complex in tropical projective coordinates. To be precise: Each row of VERTICES and LINEALITY_SPACE has a leading 1 or 0, depending on whether it is a vertex or a ray. The remaining n coordinates are interpreted as an element of Rn modulo (1,..,1). IMPORTANT NOTE: VERTICES are assumed to be normalized such that the first coordinate (i.e. column index 1) is 0. If your input is not of that form, use PROJECTIVE_VERTICES. Note that there is a convenience method thomog, which converts affine coordinates into projective coordinates.

    derived from: fan::PolyhedralComplex<Rational>
    Type Parameters
    Addition
    The tropical addition. Warning: There is NO default for this, you have to choose either Max or Min.

    Properties of Cycle

    User Methods of Cycle

    •  

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

      •  
        affine_chart (chart) → fan::PolyhedralComplex<Rational>

        This produces a version of the cycle in the coordinates of a standard tropical chart, i.e. one coordinate is set to 0. It is returned as an ordinary polyhedral complex (which can, for example, be used for visualization).

        Parameters
        Intchart
        The coordinate which should be set to 0. Indexed from 0 to AMBIENT_DIM-1 and 0 by default.
        Returns
        fan::PolyhedralComplex<Rational>
    •  

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

      Contained in extension bundled:atint.
      •  
        is_fan (allow_translations) → Bool

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

        Parameters
        Boolallow_translations
        . Optional and false by default. If true, a shifted fan is also accepted.
        Returns
        Bool
        .
    •  

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

    •  

      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.

      Contained in extension bundled:atint.
      •  
        delocalize () → Cycle

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

        Returns
        Cycle
    •  

      These methods are for visualization.

      Contained in extension bundled:atint.
      •  
        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
        Rationaldist
        . Optional, 1 by default.
        Intchart
        . 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
        IntChart
        Which affine chart to visualize, i.e. which coordinate to shift to 0. This is 0 by default.
        StringWeightLabels
        If "hidden", no weight labels are displayed. Not hidden by default.
        StringCoordLabels
        If "show", coordinate labels are displayed at vertices. Hidden by default.
        StringBoundingMode
        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.
        RationalBoundingDistance
        The distance parameter for relative bounding mode
        Matrix<Rational>BoundingBox
        The bounding parameter for absolute bounding mode
        option list:Visual::Cycle::BoundingDecorations
    •  

      These methods relate to the weights of a tropical cycle.

    Permutations of Cycle

  •  

    This is a special instance of a Cycle: It is the tropical locus of a polynomial over the tropical numbers.

    derived from: Cycle

    Properties of Hypersurface

    User Methods of Hypersurface

    •  

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

    •  

      The following methods compute topological invariants.

      •  
        GENUS ()

        The topological genus of a onedimensional hypersurface, i.e. the number of interior lattice points that occur in the dual subdivision.

    Permutations of Hypersurface

  •  

    This represents the result of the method lines_in_cubic. It contains: The tropical polynomial representing the surface, the surface itself as a Cycle and lists of lines and families of different types, each starting with LIST_...

    The object also has methods, starting with array_... that return the corresponding LIST_... as a perl array. The different (lists of) lines can be visualized nicely with visualize_in_surface.

    Contained in extension bundled:atint.

    Properties of LinesInCubic

    User Methods of LinesInCubic

  •  

    A matroid ring cycle is a tropical cycle which lies in the intersection ring of matroids fans. I.e. it is a linear combination of matroids fans (of the same dimension). Cycle sum and intersection are implemented through the combinatorics of the underlying matroids. Note that the set of loopfree nested matroids is a basis for this ring (e.g. as a Z-module). Hence every MatroidRingCycle is represented as a linear combination of nested matroids. The nested matroids are encoded via their maximal transversal presentations in MatroidRingCycle::NESTED_PRESENTATIONS. The corresponding coefficients are found in MatroidRingCycle::NESTED_COEFFICIENTS.

    Contained in extension bundled:atint.
    derived from: Cycle

    Properties of MatroidRingCycle

    User Methods of MatroidRingCycle

    •  

      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:
        • The following computes the matroid ring cycle of the matroid corresponding to the complete graph on 4 vertices. It then returns a list of the nested matroids in its basis presentation.> $r = new MatroidRingCycle<Max>(matroid::matroid_from_graph(complete(4)));> @n = $r->nested_matroids();> print $n[0]->BASES->size; 20

    Permutations of MatroidRingCycle

  •  

    A morphism is a function between cycles which is locally affine linear and respects the lattices. It is defined by a DOMAIN, which is a cycle, and values on this domain, VERTEX_VALUES and LINEALITY_VALUES, much like RationalFunction. Alternatively, it can be defined as a global affine linear function by giving a matrix and a translation vector.

    Contained in extension bundled:atint.

    Properties of Morphism

    •  

      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

    •  

      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
        Intdomain_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.
        Inttarget_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.
    •  

      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
        Morphismg
        Returns
        Morphism
        this after g
      •  
        before (g) → Morphism

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

        Parameters
        Morphismg
        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
        CycleSome
        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.
    •  

      These methods are for visualization.

      •  
        VISUAL ()

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

        Options
        StringFunctionLabels
        If set to "show", textual function representations are diplayed on cones. False by default
        option list:Visual::Cycle::FunctionDecorations
  •  

    A tropical polytope is the tropical convex hull of finitely many points in tropical projective space. It should always be defined via POINTS instead of VERTICES, as those define the combinatorics of the induced subdivision.

    Type Parameters
    Addition
    Either Min or Max. There is NO default for this, you have to choose!
    Scalar
    Rational by default. The underlying type of ordered group.

    Properties of Polytope

    User Methods of Polytope

    Permutations of Polytope

  •  

    An n-marked rational curve, identified by its SETS, i.e. its partitions of {1,...,n} and its COEFFICIENTS, i.e. the lengths of the corresponding edges.

    Contained in extension bundled:atint.

    Properties of RationalCurve

    •  

      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.

    •  

      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

    •  

      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

    •  

      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
        StringLengthLabels
        If "hidden", the edges are not labelled with their lengths. Any other text is ignored. Not set to "hidden" by default.
        option list:Visual::RationalCurve::decorations
  •  

    A rational function on a polyhedral complex. It can be described by giving its DOMAIN, a Cycle, and values on this domain - which are encoded in the properties VERTEX_VALUES and LINEALITY_VALUES. Alternatively, it can be defined by a tropical quotient of homogeneous tropical polynomials of the same degree i.e. by giving NUMERATOR and DENOMINATOR. A DOMAIN can be defined additionally (though one should take care that both functions are actually piecewise affine linear on the cells), otherwise it will be computed as the common refinement of the domains of affine linearity of the two polynomials. Note: This has nothing to do with common's RationalFunction (which is univariate). If you want to access that type or use this type from another application, be sure to prepend the appropriate namespace identifier.

    Contained in extension bundled:atint.

    Properties of RationalFunction

    •  

      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

    •  

      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>
    •  

      These methods are for visualization.

      •  
        VISUAL ()

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

        Options
        StringFunctionLabels
        If set to "show", textual function representations are diplayed on cones. False by default
        option list:Visual::Cycle::FunctionDecorations

User Functions

  •  

    These functions deal with abstract rational n-marked curves.

    Contained in extension bundled:atint.
    •  
      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
      RationalCurvecurve
      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
      RationalCurver
      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
      Intn_leaves
      The n in M_0,n. Needed to determine the dimension of the M_0,n component
      IntconeIndex
      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
      RationalCurvetype
      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
      RationalCurveAn
      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
  •  

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

    •  
      morphism_from_affine <Addition> (A, v, domain_chart, target_chart) → Morphism

      Takes a representation of a morphism on affine coordinates and converts it to projective ones.

      Contained in extension bundled:atint.
      Type Parameters
      Addition
      Min or Max
      Parameters
      Matrix<Rational>A
      . The matrix of the morphism x |-> Ax + v in affine coordinates.
      Vector<Rational>v
      . The translate of the morphism x |-> Ax + v in affine coordinates.
      Intdomain_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.
      Inttarget_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
      Morphism
    •  
      rational_fct_from_affine_denominator (p, chart) → RationalFunction

      This takes a tropical polynomial p defined on tropical affine coordinates and turns it into the rational function (1/p) on tropical homogeneous coordinates

      Contained in extension bundled:atint.
      Parameters
      Polynomial<TropicalNumber<Addition> >p
      A polynomial on affine coordinates.
      Intchart
      The index of the homogenizing coordinate. 0 by default.
      Returns
      RationalFunction
      A rational function, which on the given chart is described by (1/p).
    •  
      rational_fct_from_affine_denominator (p, chart) → RationalFunction

      Same as rational_fct_from_affine_denominator(Polynomial), except that it takes a string which it converts to a tropical polynomial using toTropicalPolynomial.

      Contained in extension bundled:atint.
      Parameters
      Stringp
      A string that will be converted to a tropical polynomial
      Intchart
      The index of the homogenizing coordinate. 0 by default.
      Returns
      RationalFunction
    •  
      rational_fct_from_affine_numerator (p, chart) → RationalFunction

      This takes a tropical polynomial defined on tropical affine coordinates and turns it into a rational function on tropical homogeneous coordinates

      Contained in extension bundled:atint.
      Parameters
      Polynomial<TropicalNumber<Addition> >p
      A polynomial on affine coordinates.
      Intchart
      The index of the homogenizing coordinate. 0 by default.
      Returns
      RationalFunction
      A rational function, which on the given chart is described by p.
    •  
      rational_fct_from_affine_numerator (p, chart) → RationalFunction

      Same as rational_fct_from_affine_numerator(Polynomial), except that it takes a string which it converts to a tropical polynomial using toTropicalPolynomial.

      Contained in extension bundled:atint.
      Parameters
      Stringp
      A string that will be converted to a tropical polynomial
      Intchart
      The index of the homogenizing coordinate. 0 by default.
      Returns
      RationalFunction
    •  
      tdehomog (A, chart, has_leading_coordinate) → Matrix<Rational>

      This is the inverse operation of thomog. It assumes a list of rays and vertices is given in tropical projective coordinates and returns a conversion into affine coordinates.

      Parameters
      Matrix<Rational>A
      The matrix. Can also be given as an anonymous array.
      Intchart
      Optional. Indicates which coordinate should be shifted to 0. If there is a leading coordinate, the first column of the matrix will remain untouched and the subsequent ones are numbered from 0. The default value for this is 0.
      Boolhas_leading_coordinate
      Whether the matrix has a leading 1/0 to indicate whether a row is a vertex or a ray. In that case, this coordinate is not touched. This is true by default.
      Returns
      Matrix<Rational>
    •  
      thomog (A, chart, has_leading_coordinate) → Matrix<Rational>

      Converts tropical affine to tropical projective coordinates. It takes a matrix of row vectors in Rn-1 and identifies the latter with Rn mod (1,..,1) by assuming a certain coordinate has been set to 0. I.e. it will return the matrix with a 0 column inserted at the position indicated by chart

      Parameters
      Matrix<Rational>A
      The matrix. Can also be given as an anonymous array [[..],[..],..]
      Intchart
      Optional. Indicates, which coordinate of Rn mod (1,..,1) should be set to 0 to identify it with Rn-1. Note that if there is a leading coordinate, the first column is supposed to contain the 1/0-coordinate indicating whether a row is a vertex or a ray and the remaining coordinates are then labelled 0,..,n-1. This option is 0 by default.
      Boolhas_leading_coordinate
      Whether the matrix has a leading 1/0 to indicate whether a row is a vertex or a ray. In that case, this coordinate is not touched. This is true by default.
      Returns
      Matrix<Rational>
  •  

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

    Contained in extension bundled:atint.
    •  
      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
      Cyclecycles
      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
      Boolcheck_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
      BooltestFan
      (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
      CycleA
      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
      Cyclecycle
      An arbitrary Cycle
      Cyclecontainer
      A cycle containing the first one (as a set) Doesn't need to have any weights and its tropical addition is irrelevant.
      BoolforceLatticeComputation
      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
      Cyclecomplex
      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
      CycleA
      CycleB
      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.
      Intk
      The dimension of the skeleton that should be computed
      BoolpreserveRays
      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
  •  

    These functions deal with the conversion of tropical objects between Min and Max.

    •  
      dual_addition_version (polytope, strong_conversion) → Polytope

      This function takes a tropical polytope and returns a tropical polytope that uses the opposite tropical addition. By default, the signs of the POINTS are inverted.

      Parameters
      Polytope<Addition,Scalar>polytope
      Boolstrong_conversion
      This is optional and TRUE by default. It indicates, whether the signs of the vertices should be inverted.
      Returns
      Polytope
    •  
      dual_addition_version (cycle, strong_conversion) → Cycle

      This function takes a tropical cycle and returns a tropical cycle that uses the opposite tropical addition. By default, the signs of the vertices are inverted.

      Parameters
      Cycle<Addition>cycle
      Boolstrong_conversion
      This is optional and TRUE by default. It indicates, whether the signs of the vertices should be inverted.
      Returns
      Cycle
    •  
      dual_addition_version (number, strong_conversion) → TropicalNumber

      This function takes a tropical number and returns a tropical number that uses the opposite tropical addition. By default, the sign is inverted.

      Parameters
      TropicalNumber<Addition,Scalar>number
      Boolstrong_conversion
      This is optional and TRUE by default. It indicates, whether the sign of the number should be inverted.
      Returns
      TropicalNumber
    •  
      dual_addition_version (vector, strong_conversion) → Vector<TropicalNumber>

      This function takes a vector of tropical numbers and returns a vector that uses the opposite tropical addition. By default, the signs of the entries are inverted.

      Parameters
      Vector<TropicalNumber<Addition,Scalar> >vector
      Boolstrong_conversion
      This is optional and TRUE by default. It indicates, whether the signs of the entries should be inverted.
      Returns
      Vector<TropicalNumber>
    •  
      dual_addition_version (matrix, strong_conversion) → Matrix<TropicalNumber>

      This function takes a matrix of tropical numbers and returns a matrix that uses the opposite tropical addition. By default, the signs of the entries are inverted.

      Parameters
      Matrix<TropicalNumber<Addition,Scalar> >matrix
      Boolstrong_conversion
      This is optional and TRUE by default. It indicates, whether the signs of the entries should be inverted.
      Returns
      Matrix<TropicalNumber>
    •  
      dual_addition_version (ring) → Ring<TropicalNumber>

      This function takes a ring over the tropical numbers and returns a ring that uses the opposite tropical addition. Variable names are preserved

    •  
      dual_addition_version (polynomial, strong_conversion) → Polynomial<TropicalNumber>

      This function takes a tropical polynomial and returns a tropical polynomial that uses the opposite tropical addition. By default, the signs of the coefficients are inverted.

      Parameters
      Polynomial<TropicalNumber<Addition,Scalar> >polynomial
      Boolstrong_conversion
      This is optional and TRUE by default. It indicates, whether the signs of the coefficients should be inverted.
      Returns
      Polynomial<TropicalNumber>
    •  
      dual_addition_version (M) → MatroidRingCycle

      Takes a MatroidRingCycle and converts it to the dual tropical addition

      Contained in extension bundled:atint.
  •  

    These functions create specific morphisms and functions.

    Contained in extension bundled:atint.
    •  
      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
      Intn
      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
      Intn
      The dimension of the larger torus
      Intm
      The dimension of the smaller torus
      Returns
      Morphism
      The projection map
  •  

    These functions are special +++ for creating special tropical cycles.

    Contained in extension bundled:atint.
    •  
      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
      Integerweight
      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
      Intn
      The (projective) ambient dimension
      Intk
      The (projective) dimension of the variety.
      Rationalh
      Optional, 1 by default. It is a nonnegative number, describing the height of the one interior lattice point of the cross polytope.
      Integerweight
      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
      Intambient_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
      Rationala
      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.
      IntegerThe
      (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::MatroidM
      A matroid
      Intscale
      An optional linear coefficient. The resulting cycle will be scale*B(M) in the ring of matroids.
      Returns
      MatroidRingCycle<Addition>

      Example:
      • Computes the matroid ring cycle of the uniform matroid of rank 2 on 3 elements in two equivalent ways:> $r = matroid_ring_cycle<Max>(matroid::uniform_matroid(2,3));> $s = new MatroidRingCycle<Max>(matroid::uniform_matroid(2,3));
    •  
      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.
      Intchart
      On which chart the cones should be orthants, 0 by default.
      Integerweight
      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
      Intn
      The tropical projective dimension.
      Integerw
      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
      Intn
      The ambient (projective) dimension.
      Intk
      The (projective dimension of the fan.
      Integerweight
      The global weight of the cycle. 1 by default.
      Returns
      Cycle
      A tropical linear space.
  •  

    These functions test cycles for degeneracy, i.e. whether a cycle is the empty cycle

    Contained in extension bundled:atint.
    •  
      is_empty ()

      This tests wheter a cycle is the empty cycle.

  •  

    These functions deal with the computation of divisors

    Contained in extension bundled:atint.
    •  
      divisor (C, F) → Cycle

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

      Parameters
      CycleC
      A tropical cycle
      RationalFunctionF
      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
      CycleC
      A tropical cycle
      RationalFunctionF
      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
      IncidenceMatrixcones
      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
  •  

    These functions are wrappers for gfan functions.

    •  
      gfan_tropicalbruteforce (I) → fan::SymmetricFan

      Calls gfan_tropicalbruteforce for a homogeneous ideal. If the ideal contains a monomial, gfan will return an empty object and the xslt parsing fails. We do not catch this for you.

      Parameters
      ideal::IdealI
      homogeneous ideal
      Returns
      fan::SymmetricFan
    •  
      gfan_tropicalhypersurface (p) → Cycle<Max>

      Calls gfan_tropicalhypersurface for a single polynomial. If the polynomial is a monomial, gfan will return an empty object and the xslt parsing fails. We do not catch this for you.

      Parameters
      Polynomial<Rational>p
      homogeneous polynomial
      Returns
      Cycle<Max>
    •  
      gfan_tropicalintersection (I) → Cycle<Max>

      Calls gfan_tropicalintersection for a homogeneous ideal.

      Parameters
      ideal::IdealI
      homogeneous ideal
      Returns
      Cycle<Max>
      most likely not balanced
    •  
      gfan_tropicalvariety_of_prime (I) → Cycle<Max>

      Calls gfan_tropicalstartingcone | gfan_tropicaltraverse for a homogeneous prime ideal. If the ideal contains a monomial, gfan will return an empty object and the xslt parsing fails. We do not catch this for you.

      Parameters
      ideal::IdealI
      homogeneous prime ideal
      Returns
      Cycle<Max>
  •  

    These functions deal with the creation and study of tropical Hurwitz cycles.

    Contained in extension bundled:atint.
    •  
      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
      Intk
      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
      BoolVerbose
      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
      Intk
      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
      Intk
      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
      BoolVerbose
      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
      Intk
      Vector<Int>degree
      RationalCurvelocal_curve
      Options
      BoolVerbose
      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
      Intk
      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
      BoolVerbose
      If true, the function outputs some progress information. True by default.
      Returns
      Cycle
      A subdivision of M_0,n
  •  

    These are general functions related to intersection theory.

    Contained in extension bundled:atint.
    •  
      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
      CycleA
      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
      CycleX
      A tropical cycle
      CycleY
      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
      CycleX
      A tropical cycle
      CycleY
      A tropical cycle, living in the same space as X
      Boolensure_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
      Morphismm
      A morphism.
      RationalFunctionr
      A rational function.
      Returns
      RationalFunction
      The pullback m*r.
  •  

    These functions deal with finding rational functions to given divisors.

    Contained in extension bundled:atint.
    •  
      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
  •  

    These functions deal with lattices (meaning free abelian, finitely generated groups).

    Contained in extension bundled:atint.
    •  
      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
      Intmax_arg
      The upper bound for the random integers
      Intn
      The number of integers to be created
      Returns
      Array<Integer>
  •  

    These functions deal with the computation and representation of (families of) lines in surfaces.

    Contained in extension bundled:atint.
    •  
      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>
  •  

    These functions 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.

    Contained in extension bundled:atint.
    •  
      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
      Intface
      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
      IncidenceMatrixcones
      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
      Intray
      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
      Inti
      The index of a vertex in VERTICES, which should not be a ray
      Returns
      Cycle<Addition>
      The Star of C at the vertex
  •  

    These functions deal with the arithmetics of MatroidRingCycle objects.

    Contained in extension bundled:atint.
    •  
      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
      MatroidRingCycleL
      A list of matroid ring cycles.
      Returns
      Matrix<Rational>
      A matrix representation of the linear space spanned by L

      Example:
      • The following computes 4 cycles of matroids of rank 2 on 4 elements. It then computes the corresponding linear space representation, which shows immediately that M1 + M2 = M3 + M4> $m1 = new matroid::Matroid(N_ELEMENTS=>4,BASES=>[[0,1],[0,2],[1,3],[2,3]]);> $m2 = matroid::uniform_matroid(2,4);> $m3 = new matroid::Matroid(N_ELEMENTS=>4,BASES=>[[0,1],[0,2],[0,3],[1,3],[2,3]]);> $m4 = new matroid::Matroid(N_ELEMENTS=>4,BASES=>[[0,1],[0,2],[1,2],[1,3],[2,3]]);> @r = map { matroid_ring_cycle<Min>($_)} ($m1,$m2,$m3,$m4);> print matroid_ring_linear_space(@r); -1 1 1 1 0 0 0 0 1 0 1 0
    •  
      matroid_ring_product (A, B) → MatroidRingCycle

      Computes the product of two matroid ring cycles.

    •  
      matroid_ring_sum (A, B) → MatroidRingCycle

      Computes the sum of two matroid ring cycles

    •  
      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
      Intn
      The size of the ground set.
      Returns
      MatroidRingCycle

      Example:
      • Computes the zero element of the matroid ring on 17 elements.> $z = zero_in_matroid_ring<Max>(17);> print $z->RANK; 0
  •  

    These functions deal with matroids and matroidal fans.

    Contained in extension bundled:atint.
    •  
      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::Matroidm
      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::MatroidA
      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
  •  

    These functions deal with moduli spaces of abstract or parametrized rational curves.

    Contained in extension bundled:atint.
    •  
      count_mn_cones (n, k) → Integer

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

      Parameters
      Intn
      The number of leaves. Should be >= 3
      Intk
      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
      Intn
      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
      Intn
      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)
      Inti
      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
      Intn
      The number of marked (contracted) points
      Intr
      The dimension of the target space
      Intd
      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.
      Inti
      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
      Intn
      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
      RationalCurveR ...
      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
      Intn
      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
      Intn
      The number of leaves in M_0,n
      Inti
      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
      Intn
      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
      Intn
      The number of contracted leaves
      Intd
      The number of non-contracted leaves
      Intr
      The dimension of the target space for the stable maps.
      Returns
      Cycle
      The moduli space of rational stable maps.
  •  

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

    Contained in extension bundled:atint.
    •  
      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
      Morphismf
      Morphismg
      Returns
      Morphism
  •  

    Special purpose functions.

  •  

    These functions produce a tropical hypersurface from other objects.

  •  

    These functions produce an object of type Polytope from other objects.

    •  
      cyclic <Addition> (d, n) → Polytope<Addition>

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

      Josephine Yu & Florian Block, arXiv: math.MG/0503279.
      Type Parameters
      Addition
      Min or Max.
      Parameters
      Intd
      the dimension
      Intn
      the number of generators
      Returns
      Polytope<Addition>
    •  
      hypersimplex <Addition> (d, k) → Polytope<Addition>

      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.

      Type Parameters
      Addition
      Max or Min
      Parameters
      Intd
      the dimension
      Intk
      the number of +/-1 entries
      Returns
      Polytope<Addition>
    •  
      matroid_polytope <Addition, Scalar> (m, v) → Polytope<Addition,Scalar>

      Produce the tropical matroid polytope from a matroid m. Each vertex corresponds to a basis of the matroid, the non-bases coordinates get value 0, the bases coordinates get value v, default is -orientation.

      Type Parameters
      Addition
      Min or Max
      Scalar
      coordinate type
      Parameters
      matroid::Matroidm
      Scalarv
      value for the bases
      Returns
      Polytope<Addition,Scalar>
    •  
      minkowski_sum (lambda, P, mu, Q) → Polytope<Addition,Scalar>

      Produces the tropical polytope (lambda \( \otimes \) P) \( \oplus \) (mu \( \otimes \) Q), where \( \otimes \) and \( \oplus \) are tropical scalar multiplication and tropical addition, respectively.

  •  

    These functions deal with covectors of subdivision of tropical point configurations.

    •  
      coarse_covectors (points, generators) → Matrix<int>

      This computes the coarse covector of a list of points relative to a list of generators.

      Parameters
      Matrix<TropicalNumber<Addition,Scalar> >points
      Matrix<TropicalNumber<Addition,Scalar> >generators
      Returns
      Matrix<int>
      . Rows correspond to points, columns to coordinates. Each entry encodes, how many generators contain a given point in a certain coordinate.
    •  
      coarse_covectors_of_scalar_vertices (points, maximal_cells, generators) → Matrix<int>

      Computes the coarse covectors of a list of scalar points, as described in covectors_of_scalar_vertices

      Parameters
      Matrix<Scalar>points
      IncidenceMatrixmaximal_cells
      Matrix<TropicalNumber<Addition,Scalar> >generators
      Returns
      Matrix<int>
      . Rows correspond to points, columns to coordinates. Each entry encodes, how many generators contain a given point in a certain coordinate.
    •  
      covectors (points, generators) → Array<IncidenceMatrix>

      This computes the (fine) covector of a list of points relative to a list of generators.

      Parameters
      Matrix<TropicalNumber<Addition,Scalar> >points
      Matrix<TropicalNumber<Addition,Scalar> >generators
      Returns
      Array<IncidenceMatrix>
      . Each IncidenceMatrix corresponds to a point. Rows of a matrix correspond to coordinates and columns to generators. Each row indicates, which generators contain the point in the sector corresponding to the coordinate.
    •  
      covectors_of_scalar_vertices (points, maximal_cells, generators) → Array<IncidenceMatrix>

      This computes the (fine) covector of a list of points relative to a list of generators. The points are scalar points and they are supposed to be normalized in the following sense: - All bounded vertices have a leading 1 - All unbounded vertices have a leading 0 and all nonzero entries are either +1 or -1. (but not both) Furthermore, the points make up a polyhedral complex - in particular, every maximal cell has a bounded vertex. For the bounded vertices, covectors are computed as usual. For unbounded vertices, the nonzero entries are replaced by tropical zero, the complementary entries are copied from a bounded vertex sharing a cell and then the covector is computed.

      Parameters
      Matrix<Scalar>points
      IncidenceMatrixmaximal_cells
      Matrix<TropicalNumber<Addition,Scalar> >generators
      Returns
      Array<IncidenceMatrix>
      . Each IncidenceMatrix corresponds to a point. Rows of a matrix correspond to coordinates and columns to generators. Each row indicates, which generators contain the point in the sector corresponding to the coordinate.
  •  

    These functions deal with tropical linear spaces associated to valuated matroids.

  •  

    These functions deal with general tropical arithmetic.

  •  

    These functions are for visualization.

    Contained in extension bundled:atint.
    •  
      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)

  •  

    These functions deal with the weight space of a cycle, i.e. the space of weights that make it balanced and related properties.

    Contained in extension bundled:atint.
    •  
      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
      CycleA
      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
      CycleX
      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)
  •  

    These functions relate to the weights of a tropical cycle.

    •  
      is_balanced (C)

      This computes whether a given cycle is balanced.

      Parameters
      CycleC
      The cycle for which to check balancing.$ @return Bool Whether the cycle is balanced.

Common Option Lists