Can the base ring of a polyhedron be restricted?

polyhedra

asked 2016-08-18 03:03:20 +0100

done_with_fish
273 ●8 ●17 ●33

updated 2016-08-18 04:42:33 +0100

I have a polyhedron P with rational vertices

sage: A = matrix([[1,0,0],[1,0,2],[1,1,1],[1,3/2,0]])
sage: A
[  1   0   0]
[  1   0   2]
[  1   1   1]
[  1 3/2   0]
sage: P = Polyhedron(A)
sage: P
A 2-dimensional polyhedron in QQ^3 defined as the convex hull  of 4 vertices

If we scale P by a factor of two, then we get a lattice polytope.

sage: (2*P).is_lattice_polytope()
True

However, the base ring of P is still QQ

sage: (2*P).parent()
Polyhedra in QQ^3

Since 2*P is a lattice polytope it seems like it should be possible to restrict the base ring of 2*P to ZZ. Is this possible?

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answered 2016-08-18 14:09:07 +0100

tmonteil
27423 ●31 ●204 ●517 http://wiki.sagemath.o...

updated 2016-08-18 14:12:44 +0100

I could not find any change_ring method or similar, but you can to the following by changing the ring of the vertices (note the transpose since the vertices are given as columns not rows):

sage: R  = Polyhedron((2*P).vertices_matrix().transpose().change_ring(ZZ))
sage: R
A 2-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices
sage: 2*P
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices
sage: R == 2*P
True

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Thanks for the tip. It seems like sage should automatically change the ring to ZZ whenever P is a lattice polytope.

done_with_fish ( 2016-08-18 21:31:30 +0100 )edit

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answered 2017-10-09 14:34:08 +0100

jipilab
543 ●17 ●18

updated 2017-10-09 14:34:58 +0100

The current difference between the base ring QQ and ZZ for polyhedron objects in Sage is not quite big nor quite well defined...

A polytope (a compact polyhedron) may have all vertices with integer coordinates, so that it could be called a "Lattice polytope", but its H-representation may still contain rational numbers. Of course, it is possible to multiply all coefficients in the H-representation by a certain integer factor to get an integral H-representation, but this is not automatically done in Sage.

The tickets Add .change_ring() method for polyhedra and Add .change_backend() method for polyhedra will then make it possible to apply the desired changes on the base ring and backend, if possible.

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