This page summarizes properties and methods defined for lattice polytopes in polymake. For an introductory example see here.

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. A lattice polytope is an instance of rational polytope having the property LATTICE=1. All properties and methods described below have not changed.

As facet inequalities are standardized automatically, they usually do not appear as integer vectors. polymake provides the primitive(Vector) to transform a rational or integral vector into an integral vector such that the gcd of all entries is 1 (as a return vector, the input remains unchanged). primitive(Matrix) applies this function to all rows of the matrix.

polymake can compute Groebner bases for the toric ideals associated to the lattice points in the polytope using any term order. Currently, polymake uses 4ti2 for the computation.

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 LATTICE_POINTS section. The result of the computation will be a list of vectors each representing a Groebner basis element. The Groebner basis element corresponding to a vector v in the list is a binomial g=g+-g- obtained as follows. First, one has to split the vector into its positive part v+ and negative part v-, i.e. v+-v-=v and v+, v- both non-negative. Then v+ is the exponent vector of the leading term g+ and v- the exponent vector of the trailing term g-.

GROEBNER_BASIS is a multiple object that can hold a term order and the corresponding Groebner basis. This reflects the fact that the basis depends on the choice of a term order. There are two different ways to enter a term order in polymake. You can either give a sequence of weight vectors as rows of a matrix, or you can enter one of the predefined term orders. Currently, there are three term orders defined.

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<Rational>(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,1,1]]);
polytope > print $p->LATTICE_POINTS;
1 1 0
1 0 1
1 0 0
1 1 1
polytope > $g=new GroebnerBasis("gb1", TERM_ORDER_MATRIX=>[[1,2,1,2]]);
polytope > $p->add("GROEBNER_BASIS",$g);
polytope > print $p->GROEBNER_BASIS("gb1")->BASIS;
polymake: used package 4ti2
 4ti2 -- A software package for algebraic, geometric and combinatorial problems on linear spaces.
 Copyright by 4ti2 team.
 http://www.4ti2.de/

-1 1 1 -1
polytope >$h=new GroebnerBasis("gb2", TERM_ORDER_NAME=>"deglex");
polytope > $p->add("GROEBNER_BASIS",$h);
polytope > print $p->GROEBNER_BASIS("gb2")->BASIS;
1 1 -1 -1

Note again, that the result of the Groebner basis computation has to be interpreted using the LATTICE_POINTS section of the polytope. So the binomial for the deglex order in the last computation is x1x2-x3x4, where the variables x1 and x2 correspond to the standard basis vectors, x3 to the origin and x4 to the point (1,1).