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D.13.2 polymake_lib

Computations with polytopes and fans, interface to polymake and TOPCOM
Thomas Markwig, email: keilen@mathematik.uni-kl.de
Yue Ren, email: ren@mathematik.uni-kl.de

Most procedures will not work unless polymake or topcom is installed and if so, they will only work with the operating system LINUX! For more detailed information see IMPORTANT NOTE respectively consult the help string of the procedures.

The conventions used in this library for polytopes and fans, e.g. the length and labeling of their vertices resp. rays, differs from the conventions used in polymake and thus from the conventions used in the polymake extension polymake.so of Singular. We recommend to use the newer polymake.so whenever possible.

Important note:
Even though this is a Singular library for computing polytopes and fans such as the Newton polytope or the Groebner fan of a polynomial, most of the hard computations are NOT done by Singular but by the program
- polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt (see http://www.polymake.org/),
respectively (only in the procedure triangulations) by the program
- topcom by Joerg Rambau, Universitaet Bayreuth (see http://www.rambau.wm.uni-bayreuth.de/TOPCOM/);
this library should rather be seen as an interface which allows to use a (very limited) number of options which polymake respectively topcom offers to compute with polytopes and fans and to make the results available in Singular for further computations; moreover, the user familiar with Singular does not have to learn the syntax of polymake or topcom, if the options offered here are sufficient for his purposes.
Note, though, that the procedures concerned with planar polygons are independent of both, polymake and topcom.

Procedures using polymake:

D.13.2.1 polymakePolytope  computes the vertices of a polytope using polymake
D.13.2.2 newtonPolytopeP  computes the Newton polytope of a polynomial
D.13.2.3 newtonPolytopeLP  computes the lattice points of the Newton polytope
D.13.2.4 normalFanL  computes the normal fan of a polytope
D.13.2.5 groebnerFanP  computes the Groebner fan of a polynomial
Procedures using topcom:
D.13.2.6 triangulations  computes all triangulations of a marked polytope
D.13.2.7 secondaryPolytope  computes the secondary polytope of a marked polytope
Procedures using polymake and topcom:
D.13.2.8 secondaryFan  computes the secondary fan of a marked polytope
Procedures conerned with planar polygons:
D.13.2.9 cycleLength  computes the cycleLength of cycle
D.13.2.10 splitPolygon  splits a marked polygon into vertices, facets, interior points
D.13.2.11 eta  computes the eta-vector of a triangulation
D.13.2.12 findOrientedBoundary  computes the boundary of a convex hull
D.13.2.13 cyclePoints  computes lattice points connected to some lattice point
D.13.2.14 latticeArea  computes the lattice area of a polygon
D.13.2.15 picksFormula  computes the ingrediants of Pick's formula for a polygon
D.13.2.16 ellipticNF  computes the normal form of an elliptic polygon
D.13.2.17 ellipticNFDB  displays the 16 normal forms of elliptic polygons
Procedures using libpolymake:
D.13.2.18 boundaryLatticePoints  
D.13.2.19 ehrhartPolynomialCoeff  
D.13.2.20 fVectorP  
D.13.2.21 facetVertexLatticeDistances  
D.13.2.22 facetWidth  
D.13.2.23 facetWidths  
D.13.2.24 gorensteinIndex  
D.13.2.25 gorensteinVector  
D.13.2.26 hStarVector  
D.13.2.27 hVector  
D.13.2.28 hilbertBasis  
D.13.2.29 interiorLatticePoints  
D.13.2.30 isBounded  
D.13.2.31 isCanonical  
D.13.2.32 isCompressed  
D.13.2.33 isGorenstein  
D.13.2.34 isLatticeEmpty  
D.13.2.35 isNormal  
D.13.2.36 isReflexive  
D.13.2.37 isSmooth  
D.13.2.38 isTerminal  
D.13.2.39 isVeryAmple  
D.13.2.40 latticeCodegree  
D.13.2.41 latticeDegree  
D.13.2.42 latticePoints  
D.13.2.43 latticeVolume  
D.13.2.44 maximalFace  
D.13.2.45 maximalValue  
D.13.2.46 minimalFace  
D.13.2.47 minimalValue  
D.13.2.48 minkowskiSum  
D.13.2.49 nBoundaryLatticePoints  
D.13.2.50 nHilbertBasis  
D.13.2.51 nInteriorLatticePoints  
D.13.2.52 nLatticePoints  
D.13.2.53 normalFan  
D.13.2.54 vertexAdjacencyGraph  
D.13.2.55 vertexEdgeGraph  
D.13.2.56 visual