Tutorial: Tropical arithmetics and tropical geometry

This tutorial showcases the main features of application tropical, such as

• Tropical arithmetics
• Tropical convex hull computations
• Tropical cycles and hypersurfaces.
• Tropical intersection theory

To use the full palette of tools for tropical geometry, switch to the corresponding application by typing the following in the polymake shell:

> application 'tropical';

Most objects and data types related to tropical computations have a template parameter which tells it whether Min or Max is used as tropical addition. There is no default for this, so you have to choose!

For a while now, polymake comes bundled with the extension a-tint by Simon Hampe, which specializes in (but is not limited to) tropical intersection theory, and in fact supplies a big part of polymakes functionality related to tropical geometry. You can find a non-comprehensive list of features here.

If you want to learn more you might want to check out:

• Simon Hampe & Michael Joswig: Tropical Computations in polymake, Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, pp 361-385 (2018).

You can create an element of the tropical semiring (over the rationals) simply by writing something like this:

> $a = new TropicalNumber<Max>(4);
> $b = new TropicalNumber<Min>(4);
> $c = new TropicalNumber<Min>("inf");

You can now do basic arithmetic - that is tropical addition and multiplication with these. Note that tropical numbers with different tropical additions don't mix!

> print $a * $a;
8
> print $b + $c*$b;
4
> # print $a + $b; # this won't work!

Tropical vector/matrix arithmetics also work - you can even ask for the tropical determinant!

> $m = new Matrix<TropicalNumber<Max>>([[0,1,2],[0,"-inf",3],[0,0,"-inf"]]);
> $v = new Vector<TropicalNumber<Max>>(1,1,2);
> print $m + $m;
0 1 2
0 -inf 3
0 0 -inf
> print $m * $v;
4 5 1
> print tdet($m);
4

Finally, you can also create tropical polynomials. This can be done with the special toTropicalPolynomial parser:

> $q = toTropicalPolynomial("min(2a,b+c)");
> print $q;
x_0^2 + x_1*x_2

All higher tropical objects (trop. polytopes, cycles, …) live in the tropical projective torus $ \mathbb R^n/\mathbb R\mathbf 1\cong \mathbb R^{n-1}$ (or in its natural compactification, the tropical projective space), i.e. they are defined in (tropically) homogeneous coordinates, and points are given via their unique representative whose first entry is $0$ (e.g. $(1,2,3)$ becomes $(0,1,2)$ etc.).

At the same time many properties (e.g. VERTICES) expect coordinates to be “extended” with an additional leading entry, which signals whether the vector should be interpreted as a conventional point in the torus (leading entry $ \mathbf 1$), or as a ray/direction/“point at infinity” (leading entry $ \mathbf 0$). You can read here for how this makes sense, or just think of the leading entry as purely symbolic.

Note that this contrasts to conventional real/complex/… projective coordinates, where this extension is not needed, since there the leading entry naturally indicates either an affine point ($1$) or point at infinity ($0$) - again, see here.

Example: the standard tropical max-line in the trop. 2-torus can be defined by 4 points in extended homogeneous coordinates: a point $(\mathbf 1,0,0,0)$ and 3 “rays” $(\mathbf 0,0,-1,0)$, $(\mathbf 0,0,0,-1)$, and $(\mathbf 0,0,1,1)$.

Since it can be tedious to write everything in tropically homogeneous coordinates, especially when you are already working with affine coordinates, Polymake provides the functions thomog and tdehomog to homogenize or dehomogenize (extended) coordinates, respectively.

Both have signature (matrix, chart=0, has_leading_coordinate=1), where matrix is a list of vectors that you want to (de)homogenize, has_leading coordinate is a boolean indicating whether we are dealing with extended coordinates (i.e. vectors have an additional leading coordinate), and chart is the index of the coordinate that is set to $0$ when identifying $ \mathbb R^n/\mathbb R\mathbf 1$ with $ \mathbb R^{n-1}$ (i.e. homogenizing works by simply adding a $0$ entry at index chart, dehomogenizing works by taking the representative whose entry at position chart is $0$, and then deleting that entry), all while ignoring the first entry if has_leading_coordinate is 1.

Some examples:

> print thomog([[1,3,4],[0,5,6]]);
1 0 3 4
0 0 5 6
> print thomog([[2,3,4]], 1, 0);
2 0 3 4
> print tdehomog([[1,3,4,5]]);
1 1 2
> print tdehomog([[2,3,4,5]], 1, 0);
-1 1 2

The basic object for tropical convex hull computations is Polytope (Careful: If you're not in application tropical, be sure to use the namespace identifier tropical::Polytope to distinguish it from the polytope::Polytope).

A tropical polytope should always be created via POINTS (i.e. not VERTICES), since they determine the combinatorial structure. The following creates a tropical line segment in the tropical projective plane. Note that the point (0,1,1) is not a vertex, as it is in the tropical convex hull of the other two points. However, it does play a role when computing the corresponding subdivision of the tropical projective torus into covector cells:

> $c = new Polytope<Min>(POINTS=>[[0,0,0],[0,1,1],[0,2,1]]);
> print $c->VERTICES;
0 0 0
0 2 1
> print rows_labeled($c->PSEUDOVERTICES);
0:1 0 1 1
1:0 0 -1 0
2:1 0 0 0
3:1 0 2 1
4:0 0 0 -1
5:0 0 1 1
> print $c->MAXIMAL_COVECTOR_CELLS; #Sets of PSEUDOVERTICES. They are maximal cells of the induced subdivision of the torus.
{0 1 2}
{0 1 5}
{0 2 4}
{0 3 4}
{0 3 5}
{1 2 4}
{3 4 5}
> print $c->POLYTOPE_MAXIMAL_COVECTOR_CELLS;
{0 2}
{0 3}

There are many visualization backends in polymake. Calling svg suits best for (static) Jupyter notebooks. The default is threejs, which is interactive.

> svg($c->VISUAL_SUBDIVISION);

In case you're just interested in either the subdivision of the full torus, or the polyhedral structure of the tropical polytope, the following will give you those structures as fan::PolyhedralComplex objects in affine coordinates:

> $t = $c->torus_subdivision_as_complex;
> $p = $c->polytope_subdivision_as_complex;
> print $p->VERTICES;
1 1 1
1 0 0
1 2 1
> print $p->MAXIMAL_POLYTOPES;
{0 2}
{0 3}

Note that by default, the affine chart is {x_0 = 0}. You can choose any chart {x_i = 0} by passing i as an argument to .._subdivision_as_complex.

Polymake computes the full subdivision of both the torus and the polytope as a CovectorLattice, which is just a FaceLattice with an additional map that attaches to each cell in the subdivision its covector. For more details on this data structure see the reference documentation. You can visualize the covector lattice with

> svg($c->TORUS_COVECTOR_DECOMPOSITION->VISUAL);
> svg($c->POLYTOPE_COVECTOR_DECOMPOSITION->VISUAL);

Each node in the lattice is a cell of the subdivision. The top row describes the vertices and rays of the subdivision. The bottom row is the covector of that cell with respect to the POINTS.

The main object here is Cycle, which represents a weighted and balanced, rational pure polyhedral complex in the tropical projective torus (see this section above, if you're confused by coordinates in the following examples).

A tropical cycle can be created, like a PolyhedralComplex, by specifying its vertices and maximal cells (and possibly a lineality space). The only additional data are the weights on the maximal cells.

> $x = new Cycle<Max>(PROJECTIVE_VERTICES=>[[1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1]],MAXIMAL_POLYTOPES=>[[0,1],[0,2],[0,3]],WEIGHTS=>[1,1,1]);

This creates the standard tropical (max-)line in the plane. There are two caveats to observe here:

1. The use of POINTS and INPUT_POLYTOPES is strongly discouraged. WEIGHTS always refer to MAXIMAL_POLYTOPES and the order of the latter can be different from the order in INPUT_POLYTOPES.
2. You can also define a cycle using VERTICES instead of PROJECTIVE_VERTICES. However, in that case all vertices have to be normalized such that the second coordinate (i.e. the one after the leading 0/1) is 0. I.e. in the above example, the point (0,-1,0,0) would have to be replaced by (0,0,1,1).

Entering projective coordinates can be a little tedious, since it usually just means adding a zero in front of your affine coordinates. There is a convenience function that does this for you. The following creates the excact same cycle as above:

> $x = new Cycle<Max>(VERTICES=>thomog([[1,0,0],[0,1,1],[0,-1,0],[0,0,-1]]),MAXIMAL_POLYTOPES=>[[0,1],[0,2],[0,3]],WEIGHTS=>[1,1,1]);

One can now ask for basic properties of the cycle, e.g., if it's balanced:

> print is_balanced($x);
true

Most of the time you probably won't want to input your tropical cycle directly as above. Polymake has a special data type Hypersurface for hypersurfaces of homogeneous tropical polynomials. The following creates the standard tropical min-line in the plane:

> $H = new Hypersurface<Min>(POLYNOMIAL=>toTropicalPolynomial("min(a,b,c)"));
> print $H->VERTICES;
1 0 0 0
0 0 -1 -1
0 0 1 0
0 0 0 1
> print $H->MAXIMAL_POLYTOPES;
{0 1}
{0 2}
{0 3}
> print $H->WEIGHTS;
1 1 1

Functionality related to tropical intersection theory is provided by the bundled extension atint by Simon Hampe, see here for a more extensive, but also slightly outdated documentation.

> $f = toTropicalPolynomial("max(0,x,y,z)");
> $div = divisor(projective_torus<Max>(3), rational_fct_from_affine_numerator($f));
> svg($div->VISUAL);

Here, projective_torus creates the tropical projective 3-torus (aka $ \mathbb R^3$) as a tropical fan with weight 1.

Let's create the standard tropical hyperplane in 3-space:

> $l = uniform_linear_space<Min>(3,2);

Furthermore, we compute a curve as the divisor of a rational function on $l:

> $div = divisor($l, rational_fct_from_affine_numerator(toTropicalPolynomial("min(3x+4,x-y-z,y+z+3)")));

We now want to visualize these together and since we want to put the resulting picture in a slide show, we want the picture to look a bit nicer, e.g. we want to intersect both the surface and the curve with the same bounding box (we don't want the curve to continue “beyond the surface”).

> print $l->bounding_box();
1 0 0 0
1 0 0 0
> print $div->bounding_box();
1 -1 -1 -1
1 3 0 0

The method bounding_box returns 2 opposing vertices of the smallest acceptable bounding box for the hyperplane's, respectively the curve's affine chart. Obviously the curve's box contains the one of the hyperplane, thus we may use it for both objects. The BoundingFacets option, however, expects a H-description. Along with additional options we may obtain it like that:

> $bbfacets = polytope::bounding_box_facets($div->bounding_box(), offset=>1, surplus_k=>0.1, fulldim=>0);

For zero or undefined offset it is advisable to set fulldim=>1. Visualizing several objects at the same time can be done with compose. Here we use threejs instead of svg; hence it might look less impressive in the static web version; try it yourself!

> compose($l->VISUAL(VertexStyle=>"hidden",BoundingFacets=>$bbfacets), $div->VISUAL(VertexStyle=>"hidden",EdgeColor=>"red", BoundingFacets=>$bbfacets));

Explode

Automatic explosion

Exploding speed

Transparency

Rotation

x-axis

y-axis

z-axis

Reset

Rotation speed

Display

Objects

Camera

Perspective
Orthographic

SVG

Download
New tab

Screenshot

Alternatively, we could simply specify an explicit bounding box to make the surface look more symmetric:

> #$m = new Matrix<Rational>([[1,-5,-5,-5],[1,5,5,5]]);
> #$mf = polytope::bounding_box_facets($m);
> $mf = polytope::scale(polytope::cube(3),5)->FACETS;
> compose($l->VISUAL(VertexStyle=>"hidden",BoundingFacets=>$mf), $div->VISUAL(VertexStyle=>"hidden",EdgeColor=>"red",BoundingFacets=>$mf));

Explode

Automatic explosion

Exploding speed

Transparency

Rotation

x-axis

y-axis

z-axis

Reset

Rotation speed

Display

Objects

Camera

Perspective
Orthographic

SVG

Download
New tab

Screenshot

As a special case of tropical linear spaces, one can create the space associated to the uniform matroid (with trivial valuation) as uniform_linear_space<Addition>(n,k; w=1) where

• Addition is either Min or Max
• n is the dimension of the ambient tropical projective torus.
• k is the (projective) dimension of the linear space.
• w is the (global) weight of the space. By default, this is 1. In particular, this gives the matroid fan of the uniform matroid $U_{n+1,k+1}$.

There are creation function for cycles that are supported on an affine linear space. In the special case, that a cycle is the whole projective torus, this can be created via projective_torus<Addition>(n;w=1) where n is the (projective) dimension and w is an integer weight.

If the cycle is given by a basis of the lineality space and a translation vector, one can use affine_linear_space<Addition>(matrix; vector, w =1).

Note that both the generators of the lineality space in the matrix and the translation vector should be given in tropical homogeneous coordinates, but WITHOUT a leading coordinate. If the translation vector is not given, it is equal to 0. The weight w is equal to 1 by default.

halfspace_subdivision<Addition>(a,g;w=1) creates the subdivision of the projective torus along the (true) hyperplane defined by the equation $g\cdot x = a$, where $a$ is a Rational and $g$ is a Vector. Note that the sum over all entries in $g$ must be 0 for this to be well-defined over tropical homogeneous coordinates.

A-tint uses the algorithms developped by Felipe Rincón to compute matroidal fans. The original algorithm was developed for matrix matroids but has been adapted here to also work on general matroids. You compute a matroidal fan via $m = matroid_fan<Addition>(m), where m is either a Matroid object or a matrix, whose column matroid is then considered.

The function cartesian_product computes the cartesian product of an arbitrary, comma-separated list of tropical cycles using the same tropical addition, e.g. the product of two uniform linear spaces could be computed via:

> $p = cartesian_product(uniform_linear_space<Max>(3,2), uniform_linear_space<Max>(3,1));

This is actually an operation on polyhedral complexes, since taking the skeleton forgets the weights. The function skeleton_complex(cycle,k,preservesRays) computes the k-dimensional skeleton of fan. The last argument is optional, by default false and should be set to true, if you are certain that all your rays remain in the skeleton (directional rays may disappear when taking the skeleton of a polyhedral complex). In this case, ray indices will be preserved.

You can check if two polyhedral complexes are exactly the same, i.e. have the same rays, cones and weights up to reordering and equivalence of tropical coordinates, by calling the function check_cycle_equality. Note that this function will not recognize if two complex are equal up to subdivision. The function has an optional parameter, which is TRUE by default, that determines whether weights ought to be checked for equality as well. This will be ignored if any of the complexes does not have any weights.

You can refine a weighted complex along any other complex containing it (as a set), using the function intersect_container(X,Y).

a-tint has a mechanism to allow computations locally around a given set of cones: Each Cycle has a property LOCAL_RESTRICTION. This is an IncidenceMatrix (or list of sets of ray indices). Each of these sets of ray indices is supposed to describe a cone of the complex - though not necessarily a maximal one.

If you define a complex with this property, then all computations will only be done “around” these cones. E.g. a-tint will only recognize those codimension one faces that contain one of the cones described in LOCAL_RESTRICTION.

Note: It is important, that all maximal cones of a restricted complex contain one of the LOCAL_RESTRICTION cones.

This way a bounded complex can also be made “balanced”:

> $w = new Cycle<Max>(VERTICES=>thomog([[1,0],[1,-1],[1,1]]),MAXIMAL_POLYTOPES=>[[0,1],[0,2]],WEIGHTS=>[1,1],LOCAL_RESTRICTION=>[[0]]);
> print $w->IS_BALANCED;
true

The above example describes the bounded line segment $[-1,1] \in \mathbb R$, subdivided at 0. Since we restrict ourselves locally to the vertex 0, the outer codimension one faces are not detected by a-tint and the complex is balanced.

This can improve computation speed, e.g. when one is only interested in the weight of a divisor at a certain cone.

There are some convenience functions to create a local variety from a global one:

• local_restrict(X,C): Takes a variety and localizes at a set of cones C. It removes all maximal cones that do not contain one of these cones. Here, C is an IncidenceMatrix, i.e. it is of the same form as e.g. MAXIMAL_POLYTOPES.
• local_vertex(X,r): Takes a variety and localizes at a ray/vertex, given by its row index r in VERTICES.
• local_codim_one(X,c): Takes a variety and localizes at a codimension one face, given by its row index in CODIMENSION_ONE_POLYTOPES.
• local_point(X,v): Takes a variety and refines it such that a given point v (in homog. coordinates, with leading coordinate) is in the polyhedral structure. Then it localizes at this point. This returns an error message if v is not actually in the complex.

Some examples:

> $x = uniform_linear_space<Max>(3, 2);
> $x1 = local_restrict($x, new IncidenceMatrix([[0],[2,3]])); 
>      #The tropical hyperplane, locally around the 0-th ray and the maximal cone spanned by rays 2 and 3.
>      #As a set this can be interpreted as the union of an open neighborhood of ray 0 and the interior of the maximal       
>      #cone <2,3>
> $x2 = local_vertex($x, 0); #An open neighborhood of ray 0.
> $x3 = local_codim_one($x, 0); #An open neighborhood of the codimension one face no. 0 (which is a ray)
> $x4 = local_point($x, new Vector<Rational>([1,0,1,1,0]));
>      #Refines the surface such that it contains the point (0,1,1,0), then takes an open neighborhood of that

For details see the internal documentation of these functions.

If you have a locally restricted complex and you would like to obtain the same complex without the LOCAL_RESTRICTION, you can call its user method delocalize(), e.g.

> $y = $x->delocalize();