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tutorial:apps_tropical [2015/10/05 14:49] – [Tropical cycles] 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 | ||
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* Tropical convex hull computations | * Tropical convex hull computations | ||
* Tropical cycles and hypersurfaces. | * 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 '' | ||
+ | < | ||
+ | > application ' | ||
+ | </ | ||
=== Disclaimer: Min or Max - you have to choose! === | === 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! | 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 ==== | ==== Tropical arithmetics ==== | ||
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</ | </ | ||
- | Finally, you can also create tropical polynomials. This can either | + | Finally, you can also create tropical polynomials. This can be done with the special |
< | < | ||
- | tropical > $r = new Ring< | ||
- | tropical > ($x, $y, $z) = $r-> | ||
- | tropical > $p = $x*$x + $y * $z; | ||
- | tropical > print $p; | ||
- | x0^2 + x1*x2 | ||
tropical > $q = toTropicalPolynomial(" | tropical > $q = toTropicalPolynomial(" | ||
tropical > print $q; | tropical > print $q; | ||
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==== Tropical convex hull computations ==== | ==== Tropical convex hull computations ==== | ||
- | The basic object for tropical convex hull computations is '' | + | The basic object for tropical convex hull computations is '' |
- | A tropical | + | A tropical |
< | < | ||
- | tropical > $c = new Cone< | + | tropical > $c = new Polytope< |
tropical > print $c-> | tropical > print $c-> | ||
0 0 0 | 0 0 0 | ||
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{2 3 4} | {2 3 4} | ||
{2 4 5} | {2 4 5} | ||
- | tropical > print $c->CONE_MAXIMAL_COVECTOR_CELLS; | + | tropical > print $c->POLYTOPE_MAXIMAL_COVECTOR_CELLS; |
{3 4} | {3 4} | ||
{4 5} | {4 5} | ||
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</ | </ | ||
- | In case you're just interested in either the subdivision of the full torus, or the polyhedral structure of the tropical | + | In case you're just interested in either the subdivision of the full torus, or the polyhedral structure of the tropical |
< | < | ||
tropical > $t = $c-> | tropical > $t = $c-> | ||
- | tropical > $p = $c->cone_subdivision_as_complex; | + | tropical > $p = $c->polytope_subdivision_as_complex; |
tropical > print $p-> | tropical > print $p-> | ||
1 0 0 | 1 0 0 | ||
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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 '' | 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 '' | ||
- | Polymake computes the full subdivision of both the torus and the cone as a '' | + | Polymake computes the full subdivision of both the torus and the polytope |
< | < | ||
tropical > $c-> | tropical > $c-> | ||
- | tropical > $c->CONE_COVECTOR_DECOMPOSITION-> | + | tropical > $c->POLYTOPE_COVECTOR_DECOMPOSITION-> |
</ | </ | ||
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 '' | 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 '' |