Differences

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--- tutorial:apps_tropical [2017/06/12 13:51] – [Tropical convex hull computations] typo. oroehrig
+++ user_guide:apps_tropical [2019/01/25 09:27] – ↷ Page moved from tutorial:apps_tropical to user_guide: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! ===
@@ Line 44: / 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 > $r = new Ring<TropicalNumber<Min> >(3); #Tropical polynomial ring in 3 variables
-tropical > ($x, $y, $z) = $r->variables;
-tropical > $p = $x*$x + $y * $z;
-tropical > print $p;
-x0^2 + x1*x2
 tropical > $q = toTropicalPolynomial("min(2a,b+c)");
 tropical > print $q;
@@ Line 59: / Line 59: @@
 ==== Tropical convex hull computations ====
-The basic object for tropical convex hull computations is ''Cone'' (**Careful:** If you're not in application tropical, be sure to use the namespace identifier ''tropical::Cone'' to distinguish it from the ''polytope::Cone'').
+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 cone 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):
+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 Cone<Min>(POINTS=>[[0,0,0],[0,1,1],[0,2,1]]);
+tropical > $c = new Polytope<Min>(POINTS=>[[0,0,0],[0,1,1],[0,2,1]]);
 tropical > print $c->VERTICES;
 0 0
@@ Line 82: / Line 82: @@
 {2 3 4}
 {2 4 5}
-tropical > print $c->CONE_MAXIMAL_COVECTOR_CELLS;
+tropical > print $c->POLYTOPE_MAXIMAL_COVECTOR_CELLS;
 {3 4}
 {4 5}
@@ Line 88: / Line 88: @@
 </code>
-In case you're just interested in either the subdivision of the full torus, or the polyhedral structure of the tropical cone, the following will give you those structures as ''fan::PolyhedralComplex'' objects in //affine// coordinates:
+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->cone_subdivision_as_complex;
+tropical > $p = $c->polytope_subdivision_as_complex;
 tropical > print $p->VERTICES;
 0 0
@@ Line 102: / Line 102: @@
 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 cone 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
+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;
+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''.