Tropical Varieties¶

A tropical variety is a piecewise-linear geometric object derived from a classical algebraic variety by using tropical mathematics, where the tropical semiring replaces the usual arithmetic operations.

AUTHORS:

Verrel Rievaldo Wijaya (2024-06): initial version

REFERENCES:

class sage.rings.semirings.tropical_variety.TropicalCurve(poly)[source]¶

Bases: TropicalVariety

A tropical curve in \(\RR^2\).

The tropical curve consists of line segments and half-lines, which we call edges. These edges are connected in such a way that they form a piecewise linear graph embedded in the plane. These edges meet at a vertices, where the balancing condition is satisfied. This balancing condition ensures that the sum of the outgoing slopes at each vertex is zero, reflecting the equilibrium.

EXAMPLES:

We define some tropical curves:

Sage

sage: T = TropicalSemiring(QQ, use_min=False)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = x + y + R(0)
sage: tv1 = p1.tropical_variety(); tv1
Tropical curve of 0*x + 0*y + 0
sage: tv1.components()
[[(t1, t1), [t1 >= 0], 1], [(0, t1), [t1 <= 0], 1], [(t1, 0), [t1 <= 0], 1]]
sage: tv1.plot()
Graphics object consisting of 3 graphics primitives

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = x + y + R(Integer(0))
>>> tv1 = p1.tropical_variety(); tv1
Tropical curve of 0*x + 0*y + 0
>>> tv1.components()
[[(t1, t1), [t1 >= 0], 1], [(0, t1), [t1 <= 0], 1], [(t1, 0), [t1 <= 0], 1]]
>>> tv1.plot()
Graphics object consisting of 3 graphics primitives

Sage Live

T = TropicalSemiring(QQ, use_min=False)
R.<x,y> = PolynomialRing(T)
p1 = x + y + R(0)
tv1 = p1.tropical_variety(); tv1
tv1.components()
tv1.plot()

Sage

sage: p2 = R(-2)*x^2 + R(-1)*x + R(1/2)*y + R(1/6)
sage: tv2 = p2.tropical_variety()
sage: tv2.components()
[[(1/2*t1 + 5/4, t1), [(-1/3) <= t1], 1],
 [(13/12, t1), [t1 <= (-1/3)], 2],
 [(t1, -1/3), [t1 <= (13/12)], 1]]
sage: tv2.plot()
Graphics object consisting of 4 graphics primitives

Python

>>> from sage.all import *
>>> p2 = R(-Integer(2))*x**Integer(2) + R(-Integer(1))*x + R(Integer(1)/Integer(2))*y + R(Integer(1)/Integer(6))
>>> tv2 = p2.tropical_variety()
>>> tv2.components()
[[(1/2*t1 + 5/4, t1), [(-1/3) <= t1], 1],
 [(13/12, t1), [t1 <= (-1/3)], 2],
 [(t1, -1/3), [t1 <= (13/12)], 1]]
>>> tv2.plot()
Graphics object consisting of 4 graphics primitives

Sage Live

p2 = R(-2)*x^2 + R(-1)*x + R(1/2)*y + R(1/6)
tv2 = p2.tropical_variety()
tv2.components()
tv2.plot()

When two tropical polynomials are multiplied, the tropical curve of the resulting polynomial is the union of the tropical curves of the original polynomials:

Sage

sage: p3 = p1 * p2; p3
(-2)*x^3 + (-2)*x^2*y + (-1)*x^2 + 1/2*x*y + 1/2*y^2 + 1/6*x + 1/2*y + 1/6
sage: tv3 = p3.tropical_variety()
sage: tv3.plot()
Graphics object consisting of 11 graphics primitives

Python

>>> from sage.all import *
>>> p3 = p1 * p2; p3
(-2)*x^3 + (-2)*x^2*y + (-1)*x^2 + 1/2*x*y + 1/2*y^2 + 1/6*x + 1/2*y + 1/6
>>> tv3 = p3.tropical_variety()
>>> tv3.plot()
Graphics object consisting of 11 graphics primitives

Sage Live

p3 = p1 * p2; p3
tv3 = p3.tropical_variety()
tv3.plot()

contribution()[source]¶

Return the contribution of self.

The contribution of a simple curve \(C\) is defined as the product of the normalized areas of all triangles in the corresponding dual subdivision. We just multiply positive integers attached to the trivalent vertices. The contribution of a trivalent vertex equals \(w_1w_2|\det(v_1,v_2)|\), with \(w_i\) are the weights of the adjacent edges and \(v_i\) are their weight vectors. That formula is independent of the choice made because of the balancing condition \(w_1v_1+w_2v_2+w_3v_3=0\).

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
sage: p1.tropical_variety().contribution()
1
sage: p2 = R(-1/3)*x^2 + R(1)*x*y + R(1)*y^2 + R(-1/3)*x + R(1/3)
sage: p2.tropical_variety().contribution()
16

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = R(Integer(2))*x**Integer(2) + x*y + R(Integer(2))*y**Integer(2) + x + R(-Integer(1))*y + R(Integer(3))
>>> p1.tropical_variety().contribution()
1
>>> p2 = R(-Integer(1)/Integer(3))*x**Integer(2) + R(Integer(1))*x*y + R(Integer(1))*y**Integer(2) + R(-Integer(1)/Integer(3))*x + R(Integer(1)/Integer(3))
>>> p2.tropical_variety().contribution()
16

Sage Live

T = TropicalSemiring(QQ)
R.<x,y> = PolynomialRing(T)
p1 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
p1.tropical_variety().contribution()
p2 = R(-1/3)*x^2 + R(1)*x*y + R(1)*y^2 + R(-1/3)*x + R(1/3)
p2.tropical_variety().contribution()

genus()[source]¶

Return the genus of self.

Let \(t(C)\) be the number of trivalent vertices, and let \(r(C)\) be the number of unbounded edges of \(C\). The genus of simple tropical curve \(C\) is defined by the formula:

\[g(C) = \frac{1}{2}t(C) - \frac{1}{2}r(C) + 1.\]

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = x^2 + y^2 + x*y
sage: p1.tropical_variety().genus()
1
sage: p2 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
sage: p2.tropical_variety().genus()
0

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = x**Integer(2) + y**Integer(2) + x*y
>>> p1.tropical_variety().genus()
1
>>> p2 = R(Integer(2))*x**Integer(2) + x*y + R(Integer(2))*y**Integer(2) + x + R(-Integer(1))*y + R(Integer(3))
>>> p2.tropical_variety().genus()
0

Sage Live

T = TropicalSemiring(QQ)
R.<x,y> = PolynomialRing(T)
p1 = x^2 + y^2 + x*y
p1.tropical_variety().genus()
p2 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
p2.tropical_variety().genus()

is_simple()[source]¶

Return True if self is simple and False otherwise.

A tropical curve \(C\) is called simple if each vertex is either trivalent or is locally the intersection of two line segments. Equivalently, \(C\) is simple if the corresponding subdivision consists only of triangles and parallelograms.

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = R(0) + x + y + x*y + x^2*y + x*y^2
sage: p1.tropical_variety().is_simple()
False
sage: p2 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
sage: p2.tropical_variety().is_simple()
True

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = R(Integer(0)) + x + y + x*y + x**Integer(2)*y + x*y**Integer(2)
>>> p1.tropical_variety().is_simple()
False
>>> p2 = R(Integer(2))*x**Integer(2) + x*y + R(Integer(2))*y**Integer(2) + x + R(-Integer(1))*y + R(Integer(3))
>>> p2.tropical_variety().is_simple()
True

Sage Live

T = TropicalSemiring(QQ)
R.<x,y> = PolynomialRing(T)
p1 = R(0) + x + y + x*y + x^2*y + x*y^2
p1.tropical_variety().is_simple()
p2 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
p2.tropical_variety().is_simple()

is_smooth()[source]¶

Return True if self is smooth and False otherwise.

Suppose \(C\) is a tropical curve of degree \(d\). A tropical curve \(C\) is smooth if the dual subdivision of \(C\) consists of \(d^2\) triangles each having unit area \(1/2\). This is equivalent with \(C\) having \(d^2\) vertices. These vertices are necessarily trivalent (has three adjacent edges).

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = x^2 + x + R(1)
sage: p1.tropical_variety().is_smooth()
False
sage: p2 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
sage: p2.tropical_variety().is_smooth()
True

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = x**Integer(2) + x + R(Integer(1))
>>> p1.tropical_variety().is_smooth()
False
>>> p2 = R(Integer(2))*x**Integer(2) + x*y + R(Integer(2))*y**Integer(2) + x + R(-Integer(1))*y + R(Integer(3))
>>> p2.tropical_variety().is_smooth()
True

Sage Live

T = TropicalSemiring(QQ)
R.<x,y> = PolynomialRing(T)
p1 = x^2 + x + R(1)
p1.tropical_variety().is_smooth()
p2 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
p2.tropical_variety().is_smooth()

plot()[source]¶

Return the plot of self.

Generates a visual representation of the tropical curve in cartesian coordinates. The plot shows piecewise-linear segments representing each components. The axes are centered around the vertices.

OUTPUT:

A Graphics object. The weight of the component will be written if it is greater or equal than 2. The weight is written near the vertex.

EXAMPLES:

A polynomial with only two terms will give one straight line:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<x,y> = PolynomialRing(T)
sage: (y+R(1)).tropical_variety().components()
[[(t1, 1), [-Infinity < t1, t1 < +Infinity], 1]]
sage: (y+R(1)).tropical_variety().plot()
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> (y+R(Integer(1))).tropical_variety().components()
[[(t1, 1), [-Infinity < t1, t1 < +Infinity], 1]]
>>> (y+R(Integer(1))).tropical_variety().plot()
Graphics object consisting of 1 graphics primitive

Sage Live

T = TropicalSemiring(QQ)
R.<x,y> = PolynomialRing(T)
(y+R(1)).tropical_variety().components()
(y+R(1)).tropical_variety().plot()

An intriguing and fascinating tropical curve can be obtained with a more complex tropical polynomial:

Sage

sage: p1 = R(1) + R(2)*x + R(3)*y + R(6)*x*y + R(10)*x*y^2
sage: p1.tropical_variety().components()
[[(-1, t1), [-2 <= t1], 1],
[(t1, -2), [-1 <= t1], 1],
[(t1 + 1, t1), [-4 <= t1, t1 <= -2], 1],
[(t1, -4), [t1 <= -3], 2],
[(-t1 - 7, t1), [t1 <= -4], 1]]
sage: p1.tropical_variety().plot()
Graphics object consisting of 6 graphics primitives

Python

>>> from sage.all import *
>>> p1 = R(Integer(1)) + R(Integer(2))*x + R(Integer(3))*y + R(Integer(6))*x*y + R(Integer(10))*x*y**Integer(2)
>>> p1.tropical_variety().components()
[[(-1, t1), [-2 <= t1], 1],
[(t1, -2), [-1 <= t1], 1],
[(t1 + 1, t1), [-4 <= t1, t1 <= -2], 1],
[(t1, -4), [t1 <= -3], 2],
[(-t1 - 7, t1), [t1 <= -4], 1]]
>>> p1.tropical_variety().plot()
Graphics object consisting of 6 graphics primitives

Sage Live

p1 = R(1) + R(2)*x + R(3)*y + R(6)*x*y + R(10)*x*y^2
p1.tropical_variety().components()
p1.tropical_variety().plot()

Another tropical polynomial with numerous components, resulting in a more intricate structure:

Sage

sage: p2 = (x^6 + R(4)*x^4*y^2 + R(2)*x^3*y^3 + R(3)*x^2*y^4
....:       + x*y^5 + R(7)*x^2 + R(5)*x*y + R(3)*y^2 + R(2)*x
....:       + y + R(10))
sage: p2.tropical_variety().plot() # long time
Graphics object consisting of 11 graphics primitives

Python

>>> from sage.all import *
>>> p2 = (x**Integer(6) + R(Integer(4))*x**Integer(4)*y**Integer(2) + R(Integer(2))*x**Integer(3)*y**Integer(3) + R(Integer(3))*x**Integer(2)*y**Integer(4)
...       + x*y**Integer(5) + R(Integer(7))*x**Integer(2) + R(Integer(5))*x*y + R(Integer(3))*y**Integer(2) + R(Integer(2))*x
...       + y + R(Integer(10)))
>>> p2.tropical_variety().plot() # long time
Graphics object consisting of 11 graphics primitives

Sage Live

p2 = (x^6 + R(4)*x^4*y^2 + R(2)*x^3*y^3 + R(3)*x^2*y^4
      + x*y^5 + R(7)*x^2 + R(5)*x*y + R(3)*y^2 + R(2)*x
      + y + R(10))
p2.tropical_variety().plot() # long time

Sage

sage: p3 = (R(8) + R(4)*x + R(2)*y + R(1)*x^2 + x*y + R(1)*y^2
....:       + R(2)*x^3 + x^2*y + x*y^2 + R(4)*y^3 + R(8)*x^4
....:       + R(4)*x^3*y + x^2*y^2 + R(2)*x*y^3 + y^4)
sage: p3.tropical_variety().plot() # long time
Graphics object consisting of 23 graphics primitives

Python

>>> from sage.all import *
>>> p3 = (R(Integer(8)) + R(Integer(4))*x + R(Integer(2))*y + R(Integer(1))*x**Integer(2) + x*y + R(Integer(1))*y**Integer(2)
...       + R(Integer(2))*x**Integer(3) + x**Integer(2)*y + x*y**Integer(2) + R(Integer(4))*y**Integer(3) + R(Integer(8))*x**Integer(4)
...       + R(Integer(4))*x**Integer(3)*y + x**Integer(2)*y**Integer(2) + R(Integer(2))*x*y**Integer(3) + y**Integer(4))
>>> p3.tropical_variety().plot() # long time
Graphics object consisting of 23 graphics primitives

Sage Live

p3 = (R(8) + R(4)*x + R(2)*y + R(1)*x^2 + x*y + R(1)*y^2
      + R(2)*x^3 + x^2*y + x*y^2 + R(4)*y^3 + R(8)*x^4
      + R(4)*x^3*y + x^2*y^2 + R(2)*x*y^3 + y^4)
p3.tropical_variety().plot() # long time

vertices()[source]¶

Return all vertices of self, which is the point where three or more edges intersect.

OUTPUT: a set of \((x,y)\) points

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = x + y
sage: p1.tropical_variety().vertices()
set()
sage: p2 = R(-2)*x^2 + R(-1)*x + R(1/2)*y + R(1/6)
sage: p2.tropical_variety().vertices()
{(1, -1/2), (7/6, -1/3)}

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = x + y
>>> p1.tropical_variety().vertices()
set()
>>> p2 = R(-Integer(2))*x**Integer(2) + R(-Integer(1))*x + R(Integer(1)/Integer(2))*y + R(Integer(1)/Integer(6))
>>> p2.tropical_variety().vertices()
{(1, -1/2), (7/6, -1/3)}

Sage Live

T = TropicalSemiring(QQ)
R.<x,y> = PolynomialRing(T)
p1 = x + y
p1.tropical_variety().vertices()
p2 = R(-2)*x^2 + R(-1)*x + R(1/2)*y + R(1/6)
p2.tropical_variety().vertices()

weight_vectors()[source]¶

Return the weight vectors for all vertices of self.

Weight vectors are a list of vectors associated with each vertex of the curve. Each vector corresponds to an edge emanating from that vertex and points in the direction of the edge.

Suppose \(v\) is a vertex adjacent to the edges \(e_1, \ldots, e_k\) with respective weights \(w_1, \ldots, w_k\). Every edge \(e_i\) is contained in a line (component) defined by an equation. Therefore, there exists a unique integer vector \(v_i = (\alpha, \beta)\) in the direction of \(e_i\) such that \(\gcd(\alpha, \beta)=1\). Then, each vertex \(v\) yield the vectors \(w_1 v_1, \ldots, w_k v_k\). These vectors will satisfy the following balancing condition: \(\sum_{i=1}^k w_i v_i = 0\).

OUTPUT:

A dictionary where the keys represent the vertices, and the values are lists of vectors.

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = R(-2)*x^2 + R(-1)*x + R(1/2)*y + R(1/6)
sage: p1.tropical_variety().weight_vectors()
{(1, -1/2): [(0, 1), (-1, -2), (1, 1)],
 (7/6, -1/3): [(-1, -1), (0, 1), (1, 0)]}

sage: p2 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
sage: p2.tropical_variety().weight_vectors()
{(-2, 0): [(-1, -1), (0, 1), (1, 0)],
 (-1, -3): [(-1, -1), (0, 1), (1, 0)],
 (-1, 0): [(-1, 0), (0, -1), (1, 1)],
 (3, 4): [(-1, -1), (0, 1), (1, 0)]}

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = R(-Integer(2))*x**Integer(2) + R(-Integer(1))*x + R(Integer(1)/Integer(2))*y + R(Integer(1)/Integer(6))
>>> p1.tropical_variety().weight_vectors()
{(1, -1/2): [(0, 1), (-1, -2), (1, 1)],
 (7/6, -1/3): [(-1, -1), (0, 1), (1, 0)]}

>>> p2 = R(Integer(2))*x**Integer(2) + x*y + R(Integer(2))*y**Integer(2) + x + R(-Integer(1))*y + R(Integer(3))
>>> p2.tropical_variety().weight_vectors()
{(-2, 0): [(-1, -1), (0, 1), (1, 0)],
 (-1, -3): [(-1, -1), (0, 1), (1, 0)],
 (-1, 0): [(-1, 0), (0, -1), (1, 1)],
 (3, 4): [(-1, -1), (0, 1), (1, 0)]}

Sage Live

T = TropicalSemiring(QQ)
R.<x,y> = PolynomialRing(T)
p1 = R(-2)*x^2 + R(-1)*x + R(1/2)*y + R(1/6)
p1.tropical_variety().weight_vectors()
p2 = R(2)*x^2 + x*y + R(2)*y^2 + x + R(-1)*y + R(3)
p2.tropical_variety().weight_vectors()

class sage.rings.semirings.tropical_variety.TropicalSurface(poly)[source]¶

Bases: TropicalVariety

A tropical surface in \(\RR^3\).

The tropical surface consists of planar regions and facets, which we can call cells. These cells are connected in such a way that they form a piecewise linear structure embedded in three-dimensional space. These cells meet along edges, where the balancing condition is satisfied. This balancing condition ensures that the sum of the outgoing normal vectors at each edge is zero, reflecting the equilibrium.

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ, use_min=False)
sage: R.<x,y,z> = PolynomialRing(T)
sage: p1 = x + y + z + R(0)
sage: tv = p1.tropical_variety(); tv
Tropical surface of 0*x + 0*y + 0*z + 0
sage: tv.components()
[[(t1, t1, t2), [t2 <= t1, 0 <= t1], 1],
[(t1, t2, t1), [max(0, t2) <= t1], 1],
[(0, t1, t2), [t2 <= 0, t1 <= 0], 1],
[(t1, t2, t2), [max(0, t1) <= t2], 1],
[(t1, 0, t2), [t2 <= 0, t1 <= 0], 1],
[(t1, t2, 0), [t1 <= 0, t2 <= 0], 1]]

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p1 = x + y + z + R(Integer(0))
>>> tv = p1.tropical_variety(); tv
Tropical surface of 0*x + 0*y + 0*z + 0
>>> tv.components()
[[(t1, t1, t2), [t2 <= t1, 0 <= t1], 1],
[(t1, t2, t1), [max(0, t2) <= t1], 1],
[(0, t1, t2), [t2 <= 0, t1 <= 0], 1],
[(t1, t2, t2), [max(0, t1) <= t2], 1],
[(t1, 0, t2), [t2 <= 0, t1 <= 0], 1],
[(t1, t2, 0), [t1 <= 0, t2 <= 0], 1]]

Sage Live

T = TropicalSemiring(QQ, use_min=False)
R.<x,y,z> = PolynomialRing(T)
p1 = x + y + z + R(0)
tv = p1.tropical_variety(); tv
tv.components()

plot(color='random')[source]¶

Return the plot of self by constructing a polyhedron from vertices in self.polygon_vertices().

INPUT:

color – string or tuple that represent a color (default: random); random means each polygon will be assigned a different color. If instead a specific color is provided, then all polygon will be given the same color.

OUTPUT: Graphics3d Object

EXAMPLES:

A tropical surface that consist of only one cell:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<x,y,z> = PolynomialRing(T)
sage: p1 = x + z
sage: tv = p1.tropical_variety()
sage: tv.plot()
Graphics3d Object

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p1 = x + z
>>> tv = p1.tropical_variety()
>>> tv.plot()
Graphics3d Object

Sage Live

T = TropicalSemiring(QQ)
R.<x,y,z> = PolynomialRing(T)
p1 = x + z
tv = p1.tropical_variety()
tv.plot()

A tropical surface with multiple cells that exhibit complex and intriguing geometric structures:

Sage

sage: p2 = x^2 + x + y + z + R(1)
sage: tv = p2.tropical_variety()
sage: tv.plot() # long time
Graphics3d Object

Python

>>> from sage.all import *
>>> p2 = x**Integer(2) + x + y + z + R(Integer(1))
>>> tv = p2.tropical_variety()
>>> tv.plot() # long time
Graphics3d Object

Sage Live

p2 = x^2 + x + y + z + R(1)
tv = p2.tropical_variety()
tv.plot() # long time

class sage.rings.semirings.tropical_variety.TropicalVariety(poly)[source]¶

Bases: UniqueRepresentation, SageObject

A tropical variety in \(\RR^n\).

A tropical variety is defined as a corner locus of tropical polynomial function. This means it consist of all points in \(\RR^n\) for which the minimum (maximum) of the function is attained at least twice.

We represent the tropical variety as a list of lists, where the inner list consist of three parts. The first one is a parametric equations for tropical roots. The second one is the condition for parameters. The third one is the order of the corresponding component.

INPUT:

poly – a TropicalMPolynomial

ALGORITHM:

We need to determine a corner locus of this tropical polynomial function, which is all points \((x_1, x_2, \ldots, x_n)\) for which the maximum (minimum) is obtained at least twice. First, we convert each monomial to its corresponding linear function. Then for each two monomials of polynomial, we find the points where their values are equal. Since we attempt to solve the equality of two equations in \(n\) variables, the solution set will be described by \(n-1\) parameters.

Next, we need to check if the value of previous two monomials at the points in solution set is really the maximum (minimum) of function. We do this by solving the inequality of the previous monomial with all other monomials in the polynomial after substituting the parameter. This will give us the condition of parameters. Each of this condition is then combined by union operator. If this final condition is not an empty set, then it represent one component of tropical root. Then we calculate the weight of this particular component by the maximum of gcd of the numbers \(|i-k|\) and \(|j-l|\) for all pairs \((i,j)\) and \((k,l)\) such that the value of on this component is given by the corresponding monomials.

EXAMPLES:

We construct a tropical variety in \(\RR^2\), where it is called a tropical curve:

Sage

sage: T = TropicalSemiring(QQ, use_min=False)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = R(1)*x + x*y + R(0); p1
0*x*y + 1*x + 0
sage: tv = p1.tropical_variety(); tv
Tropical curve of 0*x*y + 1*x + 0
sage: tv.components()
[[(t1, 1), [t1 >= -1], 1], [(-1, t1), [t1 <= 1], 1], [(-t1, t1), [t1 >= 1], 1]]
sage: tv.vertices()
{(-1, 1)}
sage: tv.plot()
Graphics object consisting of 3 graphics primitives

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = R(Integer(1))*x + x*y + R(Integer(0)); p1
0*x*y + 1*x + 0
>>> tv = p1.tropical_variety(); tv
Tropical curve of 0*x*y + 1*x + 0
>>> tv.components()
[[(t1, 1), [t1 >= -1], 1], [(-1, t1), [t1 <= 1], 1], [(-t1, t1), [t1 >= 1], 1]]
>>> tv.vertices()
{(-1, 1)}
>>> tv.plot()
Graphics object consisting of 3 graphics primitives

Sage Live

T = TropicalSemiring(QQ, use_min=False)
R.<x,y> = PolynomialRing(T)
p1 = R(1)*x + x*y + R(0); p1
tv = p1.tropical_variety(); tv
tv.components()
tv.vertices()
tv.plot()

../../../_images/tropical_variety-10.svg

A slightly different result will be obtained if we use min-plus algebra for the base tropical semiring:

Sage

sage: T = TropicalSemiring(QQ, use_min=True)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = R(1)*x + x*y + R(0)
sage: tv = p1.tropical_variety(); tv
Tropical curve of 0*x*y + 1*x + 0
sage: tv.components()
[[(t1, 1), [t1 <= -1], 1], [(-1, t1), [t1 >= 1], 1], [(-t1, t1), [t1 <= 1], 1]]
sage: tv.plot()
Graphics object consisting of 3 graphics primitives

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=True)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = R(Integer(1))*x + x*y + R(Integer(0))
>>> tv = p1.tropical_variety(); tv
Tropical curve of 0*x*y + 1*x + 0
>>> tv.components()
[[(t1, 1), [t1 <= -1], 1], [(-1, t1), [t1 >= 1], 1], [(-t1, t1), [t1 <= 1], 1]]
>>> tv.plot()
Graphics object consisting of 3 graphics primitives

Sage Live

T = TropicalSemiring(QQ, use_min=True)
R.<x,y> = PolynomialRing(T)
p1 = R(1)*x + x*y + R(0)
tv = p1.tropical_variety(); tv
tv.components()
tv.plot()

../../../_images/tropical_variety-11.svg

Tropical variety can consist of multiple components with varying orders:

Sage

sage: T = TropicalSemiring(QQ, use_min=False)
sage: R.<x,y> = PolynomialRing(T)
sage: p1 = R(7) + T(4)*x + y + R(4)*x*y + R(3)*y^2 + R(-3)*x^2
sage: tv = p1.tropical_variety(); tv
Tropical curve of (-3)*x^2 + 4*x*y + 3*y^2 + 4*x + 0*y + 7
sage: tv.components()
[[(3, t1), [t1 <= 0], 1],
[(-t1 + 3, t1), [0 <= t1, t1 <= 2], 1],
[(t1, 2), [t1 <= 1], 2],
[(t1, 0), [3 <= t1, t1 <= 7], 1],
[(7, t1), [t1 <= 0], 1],
[(t1 - 1, t1), [2 <= t1], 1],
[(t1 + 7, t1), [0 <= t1], 1]]
sage: tv.plot()
Graphics object consisting of 8 graphics primitives

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> p1 = R(Integer(7)) + T(Integer(4))*x + y + R(Integer(4))*x*y + R(Integer(3))*y**Integer(2) + R(-Integer(3))*x**Integer(2)
>>> tv = p1.tropical_variety(); tv
Tropical curve of (-3)*x^2 + 4*x*y + 3*y^2 + 4*x + 0*y + 7
>>> tv.components()
[[(3, t1), [t1 <= 0], 1],
[(-t1 + 3, t1), [0 <= t1, t1 <= 2], 1],
[(t1, 2), [t1 <= 1], 2],
[(t1, 0), [3 <= t1, t1 <= 7], 1],
[(7, t1), [t1 <= 0], 1],
[(t1 - 1, t1), [2 <= t1], 1],
[(t1 + 7, t1), [0 <= t1], 1]]
>>> tv.plot()
Graphics object consisting of 8 graphics primitives

Sage Live

T = TropicalSemiring(QQ, use_min=False)
R.<x,y> = PolynomialRing(T)
p1 = R(7) + T(4)*x + y + R(4)*x*y + R(3)*y^2 + R(-3)*x^2
tv = p1.tropical_variety(); tv
tv.components()
tv.plot()

../../../_images/tropical_variety-12.svg

If the tropical polynomial have \(n>2\) variables, then the result will be a tropical hypersurface embedded in a real space \(\RR^n\):

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<w,x,y,z> = PolynomialRing(T)
sage: p1 = x*y + R(-1/2)*x*z + R(4)*z^2 + w*x
sage: tv = p1.tropical_variety(); tv
Tropical hypersurface of 0*w*x + 0*x*y + (-1/2)*x*z + 4*z^2
sage: tv.components()
[[(t1, t2, t3 - 1/2, t3), [t2 - 9/2 <= t3, t3 <= t1 + 1/2, t2 - 5 <= t1], 1],
[(t1, 2*t2 - t3 + 4, t3, t2), [t3 + 1/2 <= t2, t3 <= t1], 1],
[(t1, t2, t1, t3), [max(t1 + 1/2, 1/2*t1 + 1/2*t2 - 2) <= t3], 1],
[(t1, t2 + 9/2, t3, t2), [t2 <= min(t3 + 1/2, t1 + 1/2)], 1],
[(t1 - 1/2, t2, t3, t1), [t2 - 9/2 <= t1, t1 <= t3 + 1/2, t2 - 5 <= t3], 1],
[(2*t1 - t2 + 4, t2, t3, t1), [t1 <= min(1/2*t2 + 1/2*t3 - 2, t2 - 9/2)], 1]]

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('w', 'x', 'y', 'z',)); (w, x, y, z,) = R._first_ngens(4)
>>> p1 = x*y + R(-Integer(1)/Integer(2))*x*z + R(Integer(4))*z**Integer(2) + w*x
>>> tv = p1.tropical_variety(); tv
Tropical hypersurface of 0*w*x + 0*x*y + (-1/2)*x*z + 4*z^2
>>> tv.components()
[[(t1, t2, t3 - 1/2, t3), [t2 - 9/2 <= t3, t3 <= t1 + 1/2, t2 - 5 <= t1], 1],
[(t1, 2*t2 - t3 + 4, t3, t2), [t3 + 1/2 <= t2, t3 <= t1], 1],
[(t1, t2, t1, t3), [max(t1 + 1/2, 1/2*t1 + 1/2*t2 - 2) <= t3], 1],
[(t1, t2 + 9/2, t3, t2), [t2 <= min(t3 + 1/2, t1 + 1/2)], 1],
[(t1 - 1/2, t2, t3, t1), [t2 - 9/2 <= t1, t1 <= t3 + 1/2, t2 - 5 <= t3], 1],
[(2*t1 - t2 + 4, t2, t3, t1), [t1 <= min(1/2*t2 + 1/2*t3 - 2, t2 - 9/2)], 1]]

Sage Live

T = TropicalSemiring(QQ)
R.<w,x,y,z> = PolynomialRing(T)
p1 = x*y + R(-1/2)*x*z + R(4)*z^2 + w*x
tv = p1.tropical_variety(); tv
tv.components()

components()[source]¶

Return all components of self.

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<a,x,y,z> = PolynomialRing(T)
sage: tv = (a+x+y+z).tropical_variety()
sage: tv.components()
[[(t1, t1, t2, t3), [t1 <= min(t3, t2)], 1],
 [(t1, t2, t1, t3), [t1 <= t3, t1 <= t2], 1],
 [(t1, t2, t3, t1), [t1 <= min(t3, t2)], 1],
 [(t1, t2, t2, t3), [t2 <= t3, t2 <= t1], 1],
 [(t1, t2, t3, t2), [t2 <= min(t3, t1)], 1],
 [(t1, t2, t3, t3), [t3 <= min(t1, t2)], 1]]

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('a', 'x', 'y', 'z',)); (a, x, y, z,) = R._first_ngens(4)
>>> tv = (a+x+y+z).tropical_variety()
>>> tv.components()
[[(t1, t1, t2, t3), [t1 <= min(t3, t2)], 1],
 [(t1, t2, t1, t3), [t1 <= t3, t1 <= t2], 1],
 [(t1, t2, t3, t1), [t1 <= min(t3, t2)], 1],
 [(t1, t2, t2, t3), [t2 <= t3, t2 <= t1], 1],
 [(t1, t2, t3, t2), [t2 <= min(t3, t1)], 1],
 [(t1, t2, t3, t3), [t3 <= min(t1, t2)], 1]]

Sage Live

T = TropicalSemiring(QQ)
R.<a,x,y,z> = PolynomialRing(T)
tv = (a+x+y+z).tropical_variety()
tv.components()

dimension()[source]¶

Return the dimension of self.

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<a,x,y,z> = PolynomialRing(T)
sage: p1 = x*y + R(-1)*x*z
sage: p1.tropical_variety().dimension()
4

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('a', 'x', 'y', 'z',)); (a, x, y, z,) = R._first_ngens(4)
>>> p1 = x*y + R(-Integer(1))*x*z
>>> p1.tropical_variety().dimension()
4

Sage Live

T = TropicalSemiring(QQ)
R.<a,x,y,z> = PolynomialRing(T)
p1 = x*y + R(-1)*x*z
p1.tropical_variety().dimension()

number_of_components()[source]¶

Return the number of components that make up self.

EXAMPLES:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<a,x,y,z> = PolynomialRing(T)
sage: p1 = x*y*a + x*z + y^2 + a*x + y + z
sage: p1.tropical_variety().number_of_components()
13

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('a', 'x', 'y', 'z',)); (a, x, y, z,) = R._first_ngens(4)
>>> p1 = x*y*a + x*z + y**Integer(2) + a*x + y + z
>>> p1.tropical_variety().number_of_components()
13

Sage Live

T = TropicalSemiring(QQ)
R.<a,x,y,z> = PolynomialRing(T)
p1 = x*y*a + x*z + y^2 + a*x + y + z
p1.tropical_variety().number_of_components()

weight_vectors()[source]¶

Return the weight vectors for each unique intesection of components of self.

Weight vectors are a list of vectors associated with each unique intersection of the components of tropical variety. Each vector is a normal vector to a component with respect to the unique intersection lying within that component.

Assume self is a \(n\)-dimensional tropical variety. Suppose \(L\) is an intersection lying within the components \(S_1, \ldots, S_k\) with respective weights \(w_1, \ldots, w_k\). This \(L\) is a linear structure in \(\RR^{n-1}\) and has \(n-1\) direction vectors \(d_1,d_2,\ldots, d_{n-1}\). Each component \(S_1, \ldots, S_k\) has a normal vector \(n_1, \ldots, n_k\). Then, we scale each normal vector to an integer vector such that the greatest common divisor of its elements is 1.

The weight vector of a component \(S_i\) with respect to \(L\) can be found by calculating the cross product between direction vectors of \(L\) and normal vector \(n_i\).These vectors will satisfy the balancing condition \(\sum_{i=1}^k w_k v_k = 0\).

OUTPUT:

A tuple of three dictionaries. The first dictionary contains equations representing the intersections. The second dictionary contains indices of components that contains the intersection. The third dictionary contains lists of vectors.

EXAMPLES:

Weight vectors of tropical surface:

Sage

sage: T = TropicalSemiring(QQ)
sage: R.<x,y,z> = PolynomialRing(T)
sage: p = x^2 + R(-1)*y + z + R(1)
sage: tv = p.tropical_variety()
sage: tv.weight_vectors()
({0: ((1/2*u2, u2 + 1, u2), {u2 <= 1}),
  1: ((1/2, 2, u2), {1 <= u2}),
  2: ((1/2, u2, 1), {2 <= u2}),
  3: ((u1, 2, 1), {(1/2) <= u1})},
 {0: [0, 1, 3], 1: [0, 2, 4], 2: [1, 2, 5], 3: [3, 4, 5]},
 {0: [(1, 2, -5/2), (1, -5/2, 2), (-2, 1/2, 1/2)],
  1: [(-1, -2, 0), (0, 2, 0), (1, 0, 0)],
  2: [(1, 0, 2), (0, 0, -2), (-1, 0, 0)],
  3: [(0, 1, 1), (0, 0, -1), (0, -1, 0)]})

Python

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p = x**Integer(2) + R(-Integer(1))*y + z + R(Integer(1))
>>> tv = p.tropical_variety()
>>> tv.weight_vectors()
({0: ((1/2*u2, u2 + 1, u2), {u2 <= 1}),
  1: ((1/2, 2, u2), {1 <= u2}),
  2: ((1/2, u2, 1), {2 <= u2}),
  3: ((u1, 2, 1), {(1/2) <= u1})},
 {0: [0, 1, 3], 1: [0, 2, 4], 2: [1, 2, 5], 3: [3, 4, 5]},
 {0: [(1, 2, -5/2), (1, -5/2, 2), (-2, 1/2, 1/2)],
  1: [(-1, -2, 0), (0, 2, 0), (1, 0, 0)],
  2: [(1, 0, 2), (0, 0, -2), (-1, 0, 0)],
  3: [(0, 1, 1), (0, 0, -1), (0, -1, 0)]})

Sage Live

T = TropicalSemiring(QQ)
R.<x,y,z> = PolynomialRing(T)
p = x^2 + R(-1)*y + z + R(1)
tv = p.tropical_variety()
tv.weight_vectors()