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# mMixedVolume -- Compute the mixed volume of a collection of lattice polytopes

- Usage:
    `mMixedVolume W`
- Inputs:
  - W, a list, of homogeneous ideals $I_1,\ldots,I_n$ over a polynomial ring, or a list of lists of vertices of the polytopes
- Outputs:
  - an integer, mixed volume

## Description

Let $Q_1,\ldots,Q_n$ be a collection of lattice polytopes in $\mathbb{R}^n$ and let $I_1,\ldots,I_n$ be homogeneous ideals in a polynomial ring over the field of rational numbers, corresponding to the given polytopes. These ideals can be obtained using the command homIdealPolytope. The mixed volume is calculated by computing a mixed multiplicity of these ideals.

The following example computes the mixed volume of two 2-cross polytopes.

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

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

    o1 : Ideal of QQ[X ..X ]
                      1   3

    i2 : mMixedVolume {I,I}

    o2 = 4

One can also compute the mixed volume of a collection of lattice polytopes by directly entering the vertices of the polytopes. Mixed Volume in the above example can also be computed as follows.

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

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

    o3 : List

    i4 : mMixedVolume {C,C}

    o4 = 4

The following example computes the mixed volume of a 2-dimensional hypercube $H$ and a 2-cross polytope $C$.

    i5 : H = {(1,1),(1,-1),(-1,1),(-1,-1)};

    i6 : C = {(-1,0),(0,1),(1,0),(0,-1)};

    i7 : mMixedVolume {H,C}

    o7 = 8

## See also

- homIdealPolytope -- Compute the homogeneous ideal corresponding to the vertices of a lattice polytope in $\mathbb{R}^n$.
- mixedMultiplicity -- Compute a given mixed multiplicity of ideals
- mixedVolume -- computes the mixed volume of a list of polytope
- multiReesIdeal -- Compute the defining ideal of multi-Rees algebra of ideals

## Ways to use mMixedVolume:

- `mMixedVolume(List)`

## For the programmer

The object mMixedVolume is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MixedMultiplicity.m2:752:0.