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user_guide:tutorials:apps_tropical [2017/06/13 10:30] oroehrig [Tropical convex hull computations] updated to using tropical polytope instead of cones |
user_guide:tutorials:apps_tropical [2019/02/04 22:55] (current) |
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- | ====== Tutorial: Tropical arithmetics and tropical geometry ====== | + | {{page>.:latest:@FILEID@}} |

- | | + | |

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

- | * Tropical arithmetics | + | |

- | * 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 ==== | + | |

- | | + | |

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

- | <code> | + | |

- | tropical > $a = new TropicalNumber<Max>(4); | + | |

- | tropical > $b = new TropicalNumber<Min>(4); | + | |

- | tropical > $c = new TropicalNumber<Min>("inf"); | + | |

- | </code> | + | |

- | 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! | + | |

- | <code> | + | |

- | tropical > print $a * $a; | + | |

- | 8 | + | |

- | tropical > print $b + $c*$b; | + | |

- | 4 | + | |

- | tropical > #print $a + $b; This won't work! | + | |

- | </code> | + | |

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

- | <code> | + | |

- | tropical > $m = new Matrix<TropicalNumber<Max> >([[0,1,2],[0,"-inf",3],[0,0,"-inf"]]); | + | |

- | tropical > $v = new Vector<TropicalNumber<Max> >(1,1,2); | + | |

- | tropical > print $m + $m; | + | |

- | 0 1 2 | + | |

- | 0 -inf 3 | + | |

- | 0 0 -inf | + | |

- | tropical > print $m * $v; | + | |

- | 4 5 1 | + | |

- | tropical > print tdet($m); | + | |

- | 4 | + | |

- | </code> | + | |

- | | + | |

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

- | | + | |

- | <code> | + | |

- | tropical > $q = toTropicalPolynomial("min(2a,b+c)"); | + | |

- | 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 0 | + | |

- | 0 2 1 | + | |

- | tropical > print rows_labeled($c->PSEUDOVERTICES); | + | |

- | 0:0 0 1 1 | + | |

- | 1:0 0 -1 0 | + | |

- | 2:0 0 0 -1 | + | |

- | 3:1 0 0 0 | + | |

- | 4:1 0 1 1 | + | |

- | 5:1 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; | + | |

- | 1 0 0 | + | |

- | 1 1 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->CONE_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); | + | |

- | 1 | + | |

- | </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 0 -1 -1 | + | |

- | 0 0 1 0 | + | |

- | 0 0 0 1 | + | |

- | 1 0 0 0 | + | |

- | tropical > print $H->MAXIMAL_POLYTOPES; | + | |

- | {2 3} | + | |

- | {1 3} | + | |

- | {0 3} | + | |

- | tropical > print $H->WEIGHTS; | + | |

- | 1 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]]). | + | |