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===== Properties and Methods defined for lattice polytopes ===== | ===== Properties and Methods defined for lattice polytopes ===== | ||
- | This page summarizes properties and methods defined for lattice polytopes in polymake. For an introductory example see [[user_guide: | + | This page summarizes properties and methods defined for lattice polytopes in polymake. For an introductory example see [[user_guide:tutorials: |
Up to release 2.14, a lattice polytope in polymake used to be a subclass of a rational polytope, and all basic properties were derived from that. Starting with release 2.15 (or 3.0, not decided yet), there is no distinct type LatticePolytope anymore. | Up to release 2.14, a lattice polytope in polymake used to be a subclass of a rational polytope, and all basic properties were derived from that. Starting with release 2.15 (or 3.0, not decided yet), there is no distinct type LatticePolytope anymore. | ||
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| '' | | '' | ||
- | ==== Groebner Bases ==== | ||
- | |||
- | '' | ||
- | |||
- | The ideal is defined in a polynomial ring that has one variable for each lattice point in the polytope. These variables are labeled in the same order as the lattice points listed in the '' | ||
- | |||
- | '' | ||
- | |||
- | ^ Name ^ Term Order ^ | ||
- | | '' | ||
- | | '' | ||
- | | '' | ||
- | |||
- | Note however, that these names are internally converted into square matrices, so it may be more efficient to define the term order by a weight vector. | ||
- | |||
- | We explain the use with an example: | ||
- | < | ||
- | polytope > $p=new Polytope< | ||
- | polytope > print $p-> | ||
- | 1 1 0 | ||
- | 1 0 1 | ||
- | 1 0 0 | ||
- | 1 1 1 | ||
- | polytope > $g=new GroebnerBasis(" | ||
- | polytope > $p-> | ||
- | polytope > print $p-> | ||
- | polymake: used package 4ti2 | ||
- | 4ti2 -- A software package for algebraic, geometric and combinatorial problems on linear spaces. | ||
- | | ||
- | | ||
- | |||
- | -1 1 1 -1 | ||
- | polytope >$h=new GroebnerBasis(" | ||
- | polytope > $p-> | ||
- | polytope > print $p-> | ||
- | 1 1 -1 -1 | ||
- | </ | ||
- | |||
- | Note again, that the result of the Groebner basis computation has to be interpreted using the '' | ||
==== User Methods for LatticePolytope ==== | ==== User Methods for LatticePolytope ==== |