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--- tutorial:apps_tropical [2015/10/05 10:29] – created hampe
+++ user_guide:tutorials:apps_tropical [2019/01/25 09:38] – ↷ Page moved from user_guide:apps_tropical to user_guide:tutorials:apps_tropical oroehrig
@@ Line 5: / Line 5: @@
   * Tropical convex hull computations
   * Tropical cycles and hypersurfaces.
+To use the full palette of tools for tropical geometry, switch to the corresponding application by typing the following in the ''polymake'' shell:
+<code>
+> application 'tropical';
+</code>
 === Disclaimer: Min or Max - you have to choose! ===
 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!
+=== Disclaimer 2: Newest version required ===
+Most of the features described here only work in polymake version 3.0 or newer.
 ==== Tropical arithmetics ====
@@ Line 40: / Line 49: @@
 </code>
-Finally, you can also create tropical polynomials. This can either be done in the [[polynomials_tutorial|usual manner]] or with a special parser:
+Finally, you can also create tropical polynomials. This can be done with the special toTropicalPolynomial parser:
 <code>
-tropical > $r = new Ring<TropicalNumber<Min> >(3); #Tropical polynomial ring in 3 variables
+tropical > $q = toTropicalPolynomial("min(2a,b+c)");
-tropical > ($x, $y, $z) = $r->variables;
-tropical > $p = $x*$x + $y * $z;
-tropical > print $p;
-x0^2 + x1*x2
-tropical > $q = toTropicalPolynomial("min(2a,y+z)");
 tropical > print $q;
 x0^2 + x1*x2
 </code>
+==== Tropical convex hull computations ====
+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 (see the [[apps_tropical#A note on coordinates|note]] below to understand the different coordinates):
+<code>
+tropical > $c = new Polytope<Min>(POINTS=>[[0,0,0],[0,1,1],[0,2,1]]);
+tropical > print $c->VERTICES;
+0 0
+2 1
+tropical > print rows_labeled($c->PSEUDOVERTICES);
+:0 0 1 1
+:0 0 -1 0
+:0 0 0 -1
+:1 0 0 0
+:1 0 1 1
+:1 0 2 1
+tropical > print $c->MAXIMAL_COVECTOR_CELLS; #Sets of PSEUDOVERTICES. They are maximal cells of the induced subdivision of the torus.
+{0 1 4}
+{0 2 5}
+{0 4 5}
+{1 2 3}
+{1 3 4}
+{2 3 4}
+{2 4 5}
+tropical > print $c->POLYTOPE_MAXIMAL_COVECTOR_CELLS;
+{3 4}
+{4 5}
+tropical > $c->VISUAL_SUBDIVISION;
+</code>
+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:
+<code>
+tropical > $t = $c->torus_subdivision_as_complex;
+tropical > $p = $c->polytope_subdivision_as_complex;
+tropical > print $p->VERTICES;
+0 0
+1 1
+2 1
+tropical > print $p->MAXIMAL_POLYTOPES;
+{0 1}
+{1 2}
+</code>
+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 [[http://polymake.org/release_docs/snapshot/tropical.html | reference documentation]]. You can visualize the covector lattice with
+<code>
+tropical > $c->TORUS_COVECTOR_DECOMPOSITION->VISUAL;
+tropical > $c->POLYTOPE_COVECTOR_DECOMPOSITION->VISUAL;
+</code>
+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''.
+==== Tropical cycles ====
+The main object here is ''Cycle'', which represents a weighted and balanced, rational pure polyhedral complex in the tropical projective torus (see the [[apps_tropical#A note on coordinates|note]] below, 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.
+<code>
+tropical > $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]);
+</code>
+This creates the standard tropical (max-)line in the plane. There are two caveats to observe here:
+  - 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''.
+  - 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, see [[apps_tropical#A note on coordinates|note]]) 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:
+<code>
+tropical > $x = new Cycle<Max>(VERTICES=>thomog([[1,0,0],[0,1,1],[0,-1,0],[0,0,-1]]),MAXIMAL_POLYTOPES=>[[0,1],[0,2],[0,3]],WEIGHTS=>[1,1,1]);
+</code>
+One can now ask for basic properties of the cycle, e.g., if it's balanced:
+<code>
+tropical > print is_balanced($x);
+</code>
+=== Hypersurfaces ===
+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:
+<code>
+tropical > $H = new Hypersurface<Min>(POLYNOMIAL=>toTropicalPolynomial("min(a,b,c)"));
+tropical > print $H->VERTICES;
+0 -1 -1
+0 1 0
+0 0 1
+0 0 0
+tropical > print $H->MAXIMAL_POLYTOPES;
+{2 3}
+{1 3}
+{0 3}
+tropical > print $H->WEIGHTS;
+1 1
+</code>
+=== Tropical intersection theory (and much more): a-tint ===
+As of version 2.15-beta3, polymake comes bundled with the extension [[https://github.com/simonhampe/atint | a-tint]] by Simon Hampe, which specializes in (but is not limited to) tropical intersection theory. You can find a non-comprehensive list of features [[ https://github.com/simonhampe/atint/wiki/Feature-list|here]] and a user manual and some basic tutorials on [[https://github.com/simonhampe/atint/wiki/User-Manual|this page]].
+==== A note on coordinates ====
+Coordinates of tropical cones and cycles all live in //tropical projective space//, i.e. TP<sup>n-1</sup> = (T<sup>n</sup> \ 0) / (1,..,1), where T is the tropical semiring R union +/- infinity. Every element of projective space has a unique representative such that its first non-tropical-zero entry is 0 and polymake will usually normalize your input to this form.
+When describing polyhedral complexes in tropical projective space, polymake uses vectors in TP<sup>n-1</sup>, but with an additional 1 or 0 in front, indicating whether it is a vertex or a ray (see also the page on [[coordinates|homogeneous coordinates]]).