mixedVolume -- Compute the mixed volume of a collection of lattice polytopes

Synopsis

• Usage:

  mixedVolume W

• Inputs:
  • W, a list, list of homogeneous ideals $I_1,...,I_n$ over a polynomial ring, or a list of lists of vertices of the polytopes
• Optional inputs:
  • BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
  • CoefficientRing => ..., default value QQ, Choose a coefficient ring of the (output) ideal
  • DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step.
  • MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
  • PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
  • Strategy => ..., default value null, Choose a strategy for the saturation step
  • VariableBaseName => ..., default value X, Choose a base name for variables in the created ring
• Outputs:
  • an integer, mixed volume

Description

i1 : I = homIdealPolytope {(0,1,1),(1,0,1),(1,1,0),(2,1,1),(1,2,1),(1,1,2)}

             2         2         2       2       2       2
o1 = ideal (X X X , X X X , X X X , X X X , X X X , X X X )
             1 2 3   1 2 3   1 2 3   1 2 4   1 3 4   2 3 4

o1 : Ideal of QQ[X ..X ]
                  1   4

i2 : mixedVolume {I,I,I}

o2 = 8

i3 : C = {(0,1,1),(1,0,1),(1,1,0),(2,1,1),(1,2,1),(1,1,2)}

o3 = {(0, 1, 1), (1, 0, 1), (1, 1, 0), (2, 1, 1), (1, 2, 1), (1, 1, 2)}

o3 : List

i4 : mixedVolume {C,C,C}

o4 = 8