Curves

Introduction

An abstract tropical curve is a finite, loopless, mulitgraph. It is defined by an incidence matrix with vertices as columns and edges as rows.

A divisor on an abstract tropical curve is a formal linear combination of the vertices with integer coefficients. The degree of a divisor is the sum of its coefficients. A divisor is effective if all its coefficients are nonnegative.

For more definitions on the theory of divisors and linear sisyems on abstract tropical curve, we refer to Matthew Baker, Serguei Norine (2007).

The tropical Jacobian of an abstract tropical curve is the group of divisors of degree zero modulo the subgroup of principal divisors. Here a principal divisor is the divisor associated to a piecewise-linear function on the vertices by the Laplacian operator. The tropical Jacobian is a finite abelian group, with order equal to the number of maximal spanning trees in the graph. It is isomorphic to $\prod{\mathbb{Z}/n_{i}\mathbb{Z}}$, where the $n_{i}$ are the nonzero elementary divisors of the Laplacian matrix. For more details, see Matthew Baker, Serguei Norine (2007).

Construction

TropicalCurve — Method

TropicalCurve(PC::PolyhedralComplex)

Construct a tropical curve from a polyhedral complex. If the curve is embedded, vertices must are points in $\mathbb R^n$. If the curve is abstract, the polyhedral complex is empty, vertices must be 1, ..., n, and the graph is given as attribute.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4]])
3×4 IncidenceMatrix
[1, 2]
[1, 3]
[1, 4]


julia> VR = [0 0; 1 0; -1 0; 0 1]
4×2 Matrix{Int64}:
  0  0
  1  0
 -1  0
  0  1

julia> PC = PolyhedralComplex{QQFieldElem}(IM, VR)
Polyhedral complex in ambient dimension 2

julia> TC = TropicalCurve(PC)
min tropical curve in 2-dimensional Euclidean space

julia> abs_TC = TropicalCurve(IM)
Abstract min tropical curve

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DivisorOnTropicalCurve — Method

DivisorOnTropicalCurve(tc::TropicalCurve, coeffs::Vector{Int})

Construct a divisor with coefficients coeffs on an abstract tropical curve tc.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]);

julia> tc = TropicalCurve(IM)
Abstract min tropical curve

julia> coeffs = [0, 1, 1, 1];

julia> dtc = DivisorOnTropicalCurve(tc,coeffs)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [0, 1, 1, 1])

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Auxiliary functions

graph — Method

graph(tc::TropicalCurve)

Return the graph of an abstract tropical curve tc.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]);

julia> tc = TropicalCurve(IM)
Abstract min tropical curve

julia> graph(tc)
6×4 IncidenceMatrix
[1, 2]
[1, 3]
[1, 4]
[2, 3]
[2, 4]
[3, 4]

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n_nodes — Method

n_nodes(tc::TropicalCurve)

Return the number of nodes of an abstract tropical curve tc.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]);

julia> tc = TropicalCurve(IM)
Abstract min tropical curve

julia> n_nodes(tc)
4

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coefficients — Method

coefficients(dtc::DivisorOnTropicalCurve)

Construct a divisor dtc with coefficients coeffs on an abstract tropical curve.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]);

julia> tc = TropicalCurve(IM)
Abstract min tropical curve

julia> coeffs = [0, 1, 1, 1];

julia> dtc = DivisorOnTropicalCurve(tc,coeffs)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [0, 1, 1, 1])

julia> coefficients(dtc)
4-element Vector{Int64}:
 0
 1
 1
 1

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degree — Method

degree(dtc::DivisorOnTropicalCurve)

Compute the degree of a divisor dtc on an abstract tropical curve.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]);

julia> tc = TropicalCurve(IM)
Abstract min tropical curve

julia> coeffs = [0, 1, 1, 1];

julia> dtc = DivisorOnTropicalCurve(tc,coeffs)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [0, 1, 1, 1])

julia> degree(dtc)
3

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is_effective — Method

is_effective(dtc::DivisorOnTropicalCurve)

Check whether a divisor dtc on an abstract tropical curve is effective.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]);

julia> tc = TropicalCurve(IM)
Abstract min tropical curve

julia> coeffs = [0, 1, 1, 1];

julia> dtc = DivisorOnTropicalCurve(tc,coeffs)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [0, 1, 1, 1])

julia> is_effective(dtc)
true

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chip_firing_move — Method

chipfiringmove(dtc::DivisorOnTropicalCurve, position::Int)

Given a divisor dtc and vertex labelled position, compute the linearly equivalent divisor obtained by a chip firing move from the given vertex position.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]);

julia> tc = TropicalCurve(IM)
Abstract min tropical curve

julia> coeffs = [0, 1, 1, 1];

julia> dtc = DivisorOnTropicalCurve(tc,coeffs)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [0, 1, 1, 1])

julia> chip_firing_move(dtc,1)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [-3, 2, 2, 2])

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v_reduced — Method

v_reduced(dtc::DivisorOnTropicalCurve, vertex::Int)

Given a divisor dtc and vertex labelled vertex, compute the unique divisor reduced with repspect to vertex as defined in Matthew Baker, Serguei Norine (2007). The divisor dtc must have positive coefficients apart from vertex.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]);

julia> tc = TropicalCurve(IM)
Abstract min tropical curve

julia> coeffs = [0, 1, 1, 1];

julia> dtc = DivisorOnTropicalCurve(tc,coeffs)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [0, 1, 1, 1])

julia> v_reduced(dtc,1)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [3, 0, 0, 0])

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is_linearly_equivalent — Method

islinearlyequivalent(dtc1::DivisorOnTropicalCurve, dtc2::DivisorOnTropicalCurve)

Given two effective divisors dtc1 and dtc2 on the same tropical curve, check whether they are linearly equivalent.

Examples

julia> IM = IncidenceMatrix([[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]);

julia> tc = TropicalCurve(IM)
Abstract min tropical curve

julia> coeffs1 = [0, 1, 1, 1];

julia> dtc1 = DivisorOnTropicalCurve(tc,coeffs1)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [0, 1, 1, 1])

julia> coeffs2 = [3,0,0,0];

julia> dtc2 = DivisorOnTropicalCurve(tc,coeffs2)
DivisorOnTropicalCurve{min, false}(Abstract min tropical curve, [3, 0, 0, 0])

julia> is_linearly_equivalent(dtc1, dtc2)
true

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structure_tropical_jacobian — Method

structure_tropical_jacobian(TC::TropicalCurve)

Compute the elementary divisors $n_i$ of the Laplacian matrix of the tropical curve TC. The tropical Jacobian is then isomorphic to $\prod (Z/(n_i)Z)$.

Examples

julia> cg = complete_graph(5);

julia> IM1=IncidenceMatrix([[src(e), dst(e)] for e in edges(cg)])
10×5 IncidenceMatrix
[1, 2]
[1, 3]
[2, 3]
[1, 4]
[2, 4]
[3, 4]
[1, 5]
[2, 5]
[3, 5]
[4, 5]

julia> TC1 = TropicalCurve(IM1)
Abstract min tropical curve

julia> structure_tropical_jacobian(TC1)
(General) abelian group with relation matrix
[1 0 0 0; 0 5 0 0; 0 0 5 0; 0 0 0 5]

julia> cg2 = complete_graph(3);

julia> IM2=IncidenceMatrix([[src(e), dst(e)] for e in edges(cg2)])
3×3 IncidenceMatrix
[1, 2]
[1, 3]
[2, 3]

julia> TC2 = TropicalCurve(IM2)
Abstract min tropical curve

julia> structure_tropical_jacobian(TC2)
(General) abelian group with relation matrix
[1 0; 0 3]

julia> IM3 = IncidenceMatrix([[1,2],[2,3],[3,4],[4,5],[1,5]])
5×5 IncidenceMatrix
[1, 2]
[2, 3]
[3, 4]
[4, 5]
[1, 5]

julia> TC3=TropicalCurve(IM3)
Abstract min tropical curve

julia> G = structure_tropical_jacobian(TC3)
(General) abelian group with relation matrix
[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 5]

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