level: RngIntElt Default: infinity
If C lies in a toric lattice L, then return a finite sequence of elements of L that generate C as a monoid. If specified, the parameter level restricts the search for generators to that given level; this assumes that C is graded.
If C lies in a toric lattice L, then return a finite sequence of elements of L that generate C over Q_ +. The list is forced to be minimal for MinimalRGenerators, but might not be otherwise.
The set of integral lattice points in the cone C contained in the hyperplane given by the section determined by the dual lattice element H at height given by the rational number h. The intersection of H with C is required to be compact.
> L := ToricLattice(4); > B := [ L | [1,2,3,-3], [-1,0,1,0], [1,2,1,-2], [2,0,0,-1], [0,0,2,-1] ]; > L1,f := Sublattice(B); > Dimension(L1); 3 > C := Cone(B); > Points(Polytope(B)); { (1, 2, 1, -2), (2, 0, 0, -1), (-1, 0, 1, 0), (1, 2, 3, -3), (1, 0, 1, -1), (0, 0, 2, -1), (0, 1, 1, -1), (1, 1, 2, -2) } > ZGenerators(C); [ (-1, 0, 1, 0), (0, 1, 1, -1), (1, 2, 1, -2), (2, 0, 0, -1) ]Using the embedding map f, pull the cone back to the lattice span of C.
> C1 := C @@ f;In the smaller lattice, this cone is nonsingular.
> IsNonsingular(C1); trueWe can find out which points generate the cone in the smaller lattice and compare them with the generators of the original cone.
> B1 := ZGenerators(C1); > B1; [ (-1, 0, 1), (1, 1, 0), (2, 0, -1) ] > [ Index(B,Image(f,b)) : b in B1 ]; [ 2, 3, 4 ]
The polytope defined by taking the convex hull of the vertices of the polyhedron P. This is the smallest polytope which can occur as a factor in an expression of P as the Minkowski sum of a polytope and a cone.
The polyhedron defined by taking the convex hull of the integral points contained in the polyhedron P.
The tail cone of the polyhedron P.
Return true if and only if the polyhedron P is empty.
> P := Polytope([[-4,2,1],[0,3,4],[3,1,-3],[3,0,0],[2,-1,1]]);The polar polyhedron comprises dual vectors that evaluate to at least -1 on P. It is compact if and only if the origin lies in the interior of P.
> D := Polar(P); > D; 3-dimensional polyhedron D with 3-dimensional finite part with 5 vertices: ( 1/2, 1, -1), (-1/3, 1/2, 1/6), (-1/3, 1/21, -2/7), (-1/3, -3/13, -1/13), (1/67, -37/67, 11/67) and 3-dimensional infinite part given by a cone with 3 minimal generators: ( 1, 2, 0), ( 2, 9, 5), ( 7, 9, 10)Since D is infinite, there is no intrinsic to list its integral points, but one can determine whether it contains integral points at all.
> HasIntegralPoint(D); trueComputing the span of all the integral points of D is possible.
> IntegralPart(D); 3-dimensional polyhedron with 3-dimensional finite part with 6 vertices: (0, 0, 0), (0, 1, 0), (0, 2, 1), (1, 1, 1), (1, 2, -1), (2, 2, 3) and 3-dimensional infinite part given by a cone with 3 minimal generators: (1, 2, 0), (2, 9, 5), (7, 9, 10)The compact and infinite parts of D can be recovered.
> cD := CompactPart(D); > cD; 3-dimensional polytope cD with 5 vertices: ( 1/2, 1, -1), (-1/3, 1/2, 1/6), (-1/3, 1/21, -2/7), (-1/3, -3/13, -1/13), (1/67, -37/67, 11/67)Polytopes are special cases of polyhedra (those whose infinite tail cone is the zero cone, or equivalently those that are compact), and the distinction can be determined.
> IsPolytope(cD); true > IsPolytope(D); false
> C:=Cone([[0,1,0],[0,1,1],[1,1,2],[1,1,4]]); > P:=Polyhedron(C); > P; 2-dimensional polyhedron P with 1-dimensional finite part with 2 vertices: (1, 2), (1, 4) and 2-dimensional infinite part given by a cone with 2 minimal generators: (1, 0), (1, 1)
> CC:=Cone([[1,0],[1,1]]); > QQ:=Polytope([[1,2],[1,4]]); > PP:=CC + QQ; > PP; 2-dimensional polyhedron PP with 1-dimensional finite part with 2 vertices: (1, 2), (1, 4) and 2-dimensional infinite part given by a cone with 2 minimal generators: (1, 0), (1, 1)But there's a potential catch.
> PP eq P; false > Ambient(PP); 2-dimensional toric lattice Z^2 > Ambient(P); 2-dimensional toric lattice ker <1, 0, 0> > P:=ChangeAmbient(P,Ambient(PP)); > PP eq P; trueThe ambients are different simply because of the method of construction of the two polyhedra, but they can be forced into the same space.
Return true if and only if the cone C or polyhedron P has dimension equal to that of its ambient lattice.
Return true if and only if the cone C is strictly convex; that is, if there exists a hyperplane H such that C is contained on one side of H and C meets H in single point 0.
Return true if and only if the cone C or the polyhedron P is simplicial.
Return true if and only if the polyhedron P is a simplex.
Return true if and only if the polyhedron P is simple.
Return true if and only if the polyhedron P is an affine linear space.
Return true if and only if the cone C is supported at the origin of its ambient toric lattice.
The dimension of the toric cone C or polyhedron P.
The index of the sublattice generated by the minimal ( R)-generators of the cone C in its linear span.
Return true if and only if the integral polytope P is shellable; i.e. if there exists a polytope S such that each lattice point u∈P lies on the boundary of v + i * (S - v) for some 0≤i≤k, where v is the barycentre of the vertices of P. If true, also returns S, v, and k.
The ambient toric lattice of the toric cone C or polyhedron P.
Make a cone or polyhedron identical to the toric cone C or polyhedron P but lying in the toric lattice L; the identification of the existing ambient toric lattice with L is simply by identification of the two standard bases and it requires the dimensions of the two spaces to be equal.[Next][Prev] [Right] [Left] [Up] [Index] [Root]